Rheology of Frictional Grains

Rheology of frictional grains

Dissertation zur Erlangung des mathematisch-naturwissenschaftlichen Doktorgrades “Doctor rerum naturalium” der Georg-August-Universität Göttingen

im Promotionsprogramm ProPhys der Georg-August University School of Science (GAUSS)

vorgelegt von

Matthias Grob aus Mühlhausen

Göttingen, 2016 Betreuungsausschuss: • Prof. Dr. Annette Zippelius, Institut für Theoretische Physik, Georg-August-Universität Göttingen • Dr. Claus Heussinger, Institut für Theoretische Physik, Georg-August-Universität Göttingen Mitglieder der Prüfungskommission: • Referentin: Prof. Dr. Annette Zippelius, Institut für Theoretische Physik, Georg-August-Universität Göttingen • Korreferent: Prof. Dr. Reiner Kree, Institut für Theoretische Physik, Georg-August-Universität Göttingen Weitere Mitglieder der Prüfungskommission: • PD Dr. Timo Aspelmeier, Institut für Mathematische Stochastik, Georg-August-Universität Göttingen • Prof. Dr. Stephan Herminghaus, Dynamik Komplexer Fluide, Max-Planck-Institut für Dynamik und Selbstorganisation • Dr. Claus Heussinger, Institut für Theoretische Physik, Georg-August-Universität Göttingen • Prof. Cynthia A. Volkert, PhD, Institut für Materialphysik, Georg-August-Universität Göttingen

Tag der mündlichen Prüfung: 09.08.2016 Zusammenfassung

Diese Arbeit behandelt die Beschreibung des Fließens und des Blockierens von granularer Materie. Granulare Materie kann einen Verfestigungsübergang durch- laufen. Dieser wird Jamming genannt und ist maßgeblich durch vorliegende Spannungen sowie die Packungsdichte der Körner, welche das Granulat bilden, bestimmt. Die Rheologie dichter granularer Medien ist zusätzlich zu Spannung und Packungsdichte stark durch Reibung zwischen den Körnern beeinﬂusst. Wir zeigen mittels numerischer Simulationen und analytischer Betrachtungen, wie Reibung Jamming qualitativ verändert. Reibungsfreies Jamming ist ein kon- tinuierlicher Phasenübergang mit einem kritischen Punkt bei verschwindender Spannung. Reibungsbehaftetes Jamming ist ein diskontinuierlicher Phasenüber- gang mit einem kritischen Punkt bei endlicher Spannung. Der kritische Punkt bei endlicher Spannung führt zu bemerkenswertem Verhalten: Oberhalb der kritischen Packungsdichte gibt es ein Intervall an Packungsdichten, innerhalb dessen große oder kleine Spannungen zum Fließen führen, mittlere Spannungen hingegen führen zum Blockieren des Mediums. Das Fließverhalten nahe Jamming ist stark durch die Systemgröße beeinﬂusst: Es gibt eine kritische Systemgröße, oberhalb derer zeitabhängiger Fluss entsteht. Dieser zeitabhängige Fluss wird durch die Ausbildung von großskaligen Strukturen im Spannungsfeld erklärt. Sowohl die großskaligen Strukuren als auch der damit einhergehende zeitabhängige Fluss sind neuartige Phänomene im Fluss von trockenen Granulaten und durch Rei- bung hervorgerufen.

iii

Abstract

This thesis deals with the description of flow and arrest of granular matter. Granular matter can undergo a rigidity transition — called jamming — that is mainly controlled by the applied stresses and the packing fraction of the grains that constitute the medium. In addition to stress and packing fraction, interparticle friction greatly affects the rheology of granular matter. Using numerical simulations and analytical modeling, we show how novel behavior in dense flow and jamming regimes arises in the presence of friction. In particular, frictionless jamming is continuous with a critical point at zero stress. In contrast, frictional jamming is shown to exhibit a discontinuous phase transition with a critical point at finite stress. The fact that the critical point resides at finite stress gives rise to remarkable flow behavior, called reentrant flow. Explicitly, there is an interval of packing fractions above the critical packing fraction in which large or low stress leads to flow but intermediate stress jams the medium. The behavior close to jamming depends substantially on the system size, i.e., there is a critical system size above which unsteady flow emerges. Unsteady flow is rationalized by large-scale structures in the stress fields. Both, the large-scale structures and the accompanied unsteady flow, are novel phenomena regarding the flow of dry granular matter and can be attributed to interparticle friction.

Contents

I. Introduction 1

1. An introduction to granular media 3

2. Scope of the thesis 5

3. Models for grains and friction 7 3.1. Deformation of frictionless spheres ...... 7 3.2. Frictional interactions ...... 10

4. The jamming transition 13 4.1. Frictionless jamming ...... 15 4.2. Frictional jamming ...... 18

5. Shear rheology 21 5.1. Rheometry and flow curves — an overview ...... 21 5.2. Inertial flow ...... 24 5.3. Plastic flow ...... 25 5.4. Shear thickening ...... 27 5.5. Unsteady flow ...... 30

6. Active microrheology 33

II. Results 35

7. Shear rheology of frictional grains 39 7.1. Jamming of frictional particles ...... 39

vii Contents

7.2. Rheological chaos of frictional grains ...... 46 7.3. Unsteady rheology and heterogeneous ﬂow of dry frictional grains 52

8. Active microrheology of driven granular particles 65

III. Summary and discussion 73

IV. Outlook 79

V. Appendix 85

A. Details on the hydrodynamic model 87 A.1. The ﬂowing state ...... 88 A.2. The shear jammed state ...... 89

Bibliography 91

Author contribution 105

Acknowledgments 107

List of Publications 109

viii Part I.

Introduction

1. An introduction to granular media

If we measure it by tons, the material most manipulated by man is water; the second-most-manipulated is granular matter.

(de Gennes [1999])

Granular media — collections of grains — are ubiquitous in nature, daily life, and engineering disciplines. Grains constitute the seabed, dunes, beaches, and an important proportion of the earth’s soil. We use pepper, sugar, and salt, coffee beans or powder, and many other kinds of granular media in the kitchen and in our daily life. Sand is a resource of which glass, semiconductors, concrete, and ultimately buildings are made. Granular matter as a resource or consumer products is handled in industrial processes, which, in total, are estimated to be responsible for a tenth of the planet’s energy budget [Mullin, 2002]. The amount of energy for the handling of grains, i.e., for processes where granular media are deformed or transported, highlights the importance of a proper understanding of flow properties of granular matter as a complex fluid. Grains are composed of a vast number of molecules and the energy associated with motion or contact is by orders of magnitude larger than the thermal energy, kBT . Granular media are therefore considered athermal [Bi et al., 2015; Brilliantov and Pöschel, 2010]. Interactions between grains are inelastic, i.e., in a collision, energy is dissipated into internal degrees of freedom, see, e.g., [Lan- dau and Lifshitz, 1970]. Furthermore, when grains are in contact, any motion or force parallel to the contact surface is opposed by a frictional force [Johnson, 1985]. Energy scale separation, the inelastic character of collisions, and friction make granular media distinct; classical equilibrium thermodynamics and fluid dynamics are not applicable [Kadanoff, 1999].

3 1. An introduction to granular media

Granular media are reminiscent of fluids or solids but exhibit a host of unusual phenomena [Jaeger et al., 1996]. In contrast to ordinary (simple) fluids, granular fluids do not possess a constant viscosity, i.e., the ratio between shear stress and strain rate depends on the applied shear. An excessive increase in viscosity can ultimately lead to a transition into a solid disordered state with a finite shear modulus. The transition from fluid to solid is called jamming. The solid state, too, shows extraordinary behavior. Granular particles are not bound to each other by chemical bonds but by confinement in a volume. An infinitesimal extension of the confining volume can lead to the complete loss of a finite shear modulus and the initiation of a flowing state. Rheology studies flow problems of fluids which are non-Newtonian, i.e., fluids that possess a viscosity that is not constant as a function of strain rate or shear stress. The rheological properties depend on a number of aspects and parameters: the packing fraction of the particles, the particles themselves and their interaction, the geometry and boundaries of the problem, and the preparation history. In this thesis, we study these aspects, with a particular focus on the exceptional role played by friction, in a granular medium under shear.

4 2. Scope of the thesis

In the course of this thesis we deal with dense granular media under shear. In part I, we give an overview of the subject. In Chapter 3, we deal with molecular models of granular particles. Chapter 4 focuses on assemblies of grains and explains the jamming transition: a rigidity transition from a flowing state to a disordered solid state. Subsequently, Chapter 5, deals with the shear flow of densely packed particles including phenomena like shear thickening and flow heterogeneities. Chapter 6 describes an approach to probe a complex fluid by the dynamics of a probe particle. In part II, we present the results of the thesis. This part is divided in two Chapters. Chapter 7 constitutes the main results of this thesis and we investigate the rheology of frictional grains. This Chapter is divided in three sections corresponding to an article each. First, in section 7.1, we discuss the jamming transition of frictional grains in small simulation cells. Results on large systems, in which an unsteady and chaotic response emerges, are presented in section 7.2. There, we also present a simple analytical theory, which is based on coupling of hydrodynamics to a microstructure variable associated with friction and which explains the main features of our simulations. Lastly, we present results that characterize heterogeneous and unsteady flow states and connect the preceding studies in section 7.3. Chapter 8 covers a microrheology study in a dense granular medium. The nonlinear velocity-force relations of a probe particle are investigated. Part III summarizes, discusses, and connects the results with a broader context. Part IV is an outlook to future directions of research on flow of frictional grains.

3. Models for grains and friction

The interaction between grains is fundamental for the phenomena, which we associate with a complex ﬂuid, including the possibility to jam into a solid state. Grains are solid bodies and the description of the dynamics of grains in contact is a formidable task. A contact force, fij, between granular particles, i and j, of mass, m, gives rise to motion according to Newton’s equation of motion:

X mx¨i = fij, (3.1) j6=i with the coordinate, xi, of particle, i. The contact forces are governed by the dynamics of the contact and the material properties of the particles. A thorough treatment of contact mechanics including a discussion of friction is given by Johnson [1985]. Results relevant for numerical simulations are presented in Schäfer et al. [1996]. First, we give an overview of interactions of frictionless spheres in section 3.1. Frictionless spheres allow for an extensive theoretical description and a comprehensive understanding of interaction mechanisms. Second, we highlight the importance of incorporating frictional interactions in particle models and present an approach to model friction in section 3.2.

3.1. Deformation of frictionless spheres

In this thesis, we examine the dynamics of assemblies of spherical grains with numerical methods which allow for detailed resolution of positions, contacts, velocities, interparticle forces, and local stresses. In practice, spherical particles possess the advantage that a quantitative study of stress transmission is easily accessible. In laboratory experiments, disks or spheres made from photoelastic materials are used. Stress induced fringe patterns allow for direct inference of

7 3. Models for grains and friction

(ii)

(i)

Figure 3.1.: Photograph of photoelastic disks: (i) disk without fringe pattern and (ii) disk with stress originating at the contacts and leading to the fringe pattern. Disks made by Jonathan Barès, Department of Physics & Center for Nonlinear and Complex Systems, Duke Uni- versity, Durham, North Carolina 27708, USA. Photo courtesy of J. Barès. contact points and loads [Johnson, 1985]. In ﬁgure 3.1, we show an assembly of photoelastic disks. Disk (i) is not under stress while contacts on disk (ii) lead to a characteristic fringe pattern. Ideal frictionless spheres are probably the simplest form of granular grains. Frictionless spheres in contact transmit forces only along the line connecting the spheres’ centers, called the normal of a contact. Here, we consider contact interactions only, i.e., no electrostatic interactions, etc. When particles touch just slightly they deform quasistatically like an elastic solid that stores energy in the deformation. The elastic nature of the contact aims at minimizing the stored energy which results in a repulsive force. The elastic energy depends on material parameters and the geometry of the spheres: the Young’s modulus, the Poisson

8 3.1. Deformation of frictionless spheres n

Figure 3.2.: Amount of deformation of particles near the contact surface (dashed line) modeled by the overlap δ. The unit vectors, n and t, deﬁne normal and tangential directions of contact, respectively. ratio, and the radii of the particles. Frictionless elastic spheres in contact are studied in detail in Hertz [1882]; Landau and Lifshitz [1970]. In fact not all the energy is stored as potential energy in the contact. Instead, energy is irreversibly dissipated into the particles when plastic yield occurs or vibrational modes are excited, which we might hear when particles interact (see Michlmayr et al. [2012] for a review). Energy dissipation implies that the impact velocity, v0, is larger than the post-collisional velocity, v. The negative ratio between the velocities after and prior the contact, is called coeﬃcient of restitution:

v = − . (3.2) v0

In general, the coeﬃcient of restitution depends on material parameters and on the relative velocity when particles collide [Güttler et al., 2012]. The simplest model that incorporates elastic repulsion and viscous damping due to energy dissipation is the linear spring-dashpot model [Schwager and Pöschel, 2007]. The deformation, or displacement, of particles in contact is described by the overlap, δ, see ﬁgure 3.2. The contact dynamics of two spheres in contact

9 3. Models for grains and friction

with eﬀective mass, meﬀ, follows a damped harmonic oscillator, where the normal force, f (n), acts on the displacement along the normal direction, n:

(n) (n) (n) ˙ f = −k δ − η /meﬀδ n. (3.3)

The repulsive and dissipative component is governed by a spring constant, k(n), and a viscous damping constant, η(n), respectively. Analogous to Hooke’s law, the repulsive component of the force depends linearly on the amount of deformation. The dissipative component is linear with the normal component of the relative velocity of the particles, δ˙. In this particular model, one can show that the 1 coeﬃcient of restitution and the binary collision time , tn, are independent of the impact velocity. In principle, k(n) and η(n) can be tuned to meet experimental values for and tn which determine the nature of the collision completely. This simple model lacks the velocity dependent coeﬃcient of restitution and a realistic repulsion law: spherical particles in three dimensions repel with f (n) ∝ δ3/2, see [Hertz, 1882], and two-dimensional disks repel with more complicated laws, see [Gerl and Zippelius, 1999]. However, the linear spring-dashpot model proofed useful and for the sake of simplicity it is widely applied in simulations and theory.

3.2. Frictional interactions

In experiment and nature, grains are bodies with rough surfaces. The roughness implies the existence of frictional contact forces: forces which act orthogonal to normal, i.e., tangentially. Experiments on a variety of particulate media, not only granular grains, highlight the role played by friction. Particles are as diverse as PMMA spheres [Guy et al., 2015; Pan et al., 2015], corn starch [Fall et al., 2015; Jiang et al., 2015], silica spheres [Royer et al., 2016], or photoelastic disks [Bi et al., 2011]. A review on rheology with aspects on friction is given by [Denn and Morris, 2014]. Two particles in contact share a common area where the normal forces operate (see ﬁgure 3.2). When particles are frictional, rotation or tangential relative motion create tangential stress. Tangential stress results in a tangential force,

1The binary collision time is deﬁned as the duration of a contact of two particles. In general, the duration of an interaction of more than two particles deviates from the binary collision time.

10 3.2. Frictional interactions

(i) f(t) < μf(n) (ii) f(t) = μf(n) δ(t)

f(t) fext f(t) fext

f(n) f(n) Figure 3.3.: Sliding block model to illustrate Coulomb’s law of friction. The mass of the block gives rise to a normal force, f (n). Left: The block does not slide and the external force, f ext, acting on the block is balanced by the frictional force with the surface, f (t). Due to f ext, the block deforms by δ(t) = k−1f ext. Right: The block slides and the frictional force cannot balance the external force. In this depiction, µ = 1. f (t), acting along the tangential of the contact, t, and acting as a resistive force against the motion. A particle, i, with moment of inertia, I, and angular velocity,

ωi, experiences torques due to frictional contacts with particles, j:

X (t) Iω˙ i = fij × Rij, (3.4) i6=j with Rij pointing from particle i’s center to the contact with particle j. The tangential force is bound by the normal force via Coulomb’s law of friction, i.e., f (t) cannot exceed a material speciﬁc multiple of the normal force:

|f (t)| ≤ µ|f (n)|. (3.5)

The Coulomb friction coefficient, µ, is a material parameter. In a frictional contact with |f (t)| = µ|f (n)|, interfacial slip occurs and the contact is said to slide. Otherwise the contact is called non-sliding. For a pictorial description of Coulomb’s law of friction, see figure 3.3. In general, one discriminates the between a static friction coefficient, µs, when the contact is non-sliding and a dynamic friction coefficient, µd, when the contact slides. Typically, both parameters are of the same order of magnitude [Israelachvili, 2010]. Here, we set them equal:

µ = µs = µd.

11 3. Models for grains and friction

For numerical simulations of frictional grains, the model most referenced, that obeys Coulomb’s law of friction, was proposed by Cundall and Strack [1979]. In the original model, non-sliding contacts store elastic energy according to the tangential displacement, δ(t). Analogously to the normal interaction, we extend the original model by a dissipative contribution related to the relative tangential velocity at the contact, δ˙(t). A spring constant, k(t), and a viscous damping constant, η(t), determine the mechanical properties. Explicitly, if |f (t)| < µ|f (n)|:

f (t) = −k(t)δ(t) − η(t)δ˙(t) t. (3.6)

If equality holds in Coulomb’s criterion and the contact slides, we set δ(t) = µ/k(t)|f (n)| and |f (t)| = k(t)δ(t) to satisfy Coulomb’s criterion2. The model was inspired by studies of frictional contacts [Mindlin and Deresiewicz, 1953] and used in several rheology studies, see, e.g., [Silbert et al., 2001; Otsuki and Hayakawa, 2011; Chialvo et al., 2012; DeGiuli et al., 2015; Henkes et al., 2016]. The presented approach has physical ﬂaws, see Etsion [2010], but its advantages are numerical eﬃciency and conceptual simplicity. In this thesis, we consider large assemblies of particles and we do not focus on details of collision mechanics and assume that these details are not relevant for the rheology.

2It would require a study of a particular material to justify this sliding mechanism. In the literature, a number of models exist and there is no obvious consensus which is justiﬁed by a (simple) physical argument.

12 4. The jamming transition

As illustrated in figure 4.1, sparsely distributed particles in a finite volume can move freely over a finite distance, i.e., without the interaction with other particles. Densely packed grains may be blocked by contacts with their neighbors and not even an infinitesimal motion is possible. This geometric statement has important (i) (ii)

Figure 4.1.: Particles in (i) are not geometrically constrained and can move without the expense of energy. In (ii), particles are densely packed and any motion results in repulsive forces exerted by the particles at contact. implications for the mechanical behavior of the whole assembly: Volume (i) can be sheared with any shear stress but volume (ii) can sustain a ﬁnite maximum shear stress — called yield stress — without deforming. A parameter to describe the geometrical properties of an assembly of N grains with volumes Vi conﬁned in a volume V is the packing fraction:

PN V φ = i=1 i . (4.1) V

13 4. The jamming transition

fluid yield stress line Σ

jammed

φ =φ J rcp φ Figure 4.2.: Schematic jamming phase diagram for frictionless spheres. The phase and thus the mechanical response is determined by the shear stress, Σ, and the packing fraction, φ. φJ is the onset of rigidity at zero stress, Σ = 0.

When the packing fraction increases, the granular medium can jam from a flowing state, as in figure 4.1 (i), into an unordered solid state, as in figure 4.1 (ii). Figure 4.2 shows a schematic phase diagram for the system in figure 4.1, similar to the athermal plane in Liu and Nagel [1998]. The shear stress, Σ, and the packing fraction, φ, control which state is realized: fluid or jammed. A single line, the yield stress line, separates regions of flowing states from jammed states. The yield stress line touches zero stress, Σ = 0, at the jamming packing fraction, φJ, which sets the jamming point, called point J. The phase diagram in figure 4.2 serves as a high-level description of the jamming transition for a broad range of particulate media. However, the jamming transition cannot be controlled by packing fraction and stress only, but is also influenced by the preparation protocol and the mechanical properties of the particles. We take into account the particles’ roughness by considering frictional contacts, as described in Chapter 3.

14 4.1. Frictionless jamming

In section 4.1, we discuss the basic properties of the jamming transition and focus on experiments and simulations of frictionless particles1. Subsequently, we deal with jamming of frictional particles in section 4.2, where we discuss experimental and numerical results that are directly linked to frictional interactions.

4.1. Frictionless jamming

In this section, we discuss results on the jamming transition of frictionless spheres and rheological properties in the vicinity of the jamming transition. A fixed volume that is densely and randomly packed with hard spheres, see fig. 4.1(ii), cannot be deformed: The configuration is jammed. The geometric study of such packings of spheres shows that there is a largest packing fraction of randomly packed (hard) spheres, namely the random close packing, φrcp [Song et al., 2008]. The jamming packing fraction approaches random close packing in the thermodynamic limit [O’Hern et al., 2003]:

φJ → φrcp for N → ∞ with φ = const. (4.2)

The jamming transition depends on the preparation of the jammed configuration, see e.g., [Chaudhuri et al., 2010]. In particular in a finite system, φJ depends on the initial condition and is localized in a finite interval, which narrows around φrcp when the system size increases [O’Hern et al., 2002]. For systems of frictionless particles in two dimensions, the point J is accurately determined by simulations of overdamped, bidisperse, and soft particles in simple shear geometry at zero stress and packing fraction [Olsson and Teitel, 2011]:

φJ = 0.84347 ± 0.00020. (4.3)

Even though this value changes with the protocol, equation 4.3 gives a robust estimate for jamming in two dimensions. Depending on the protocol, Chaudhuri et al. [2010] observed a range of packing fractions, (φl, φu), in which jamming is possible with a relative size of (φu −φl)/φu ≈ 0.023% in the thermodynamic limit in three dimensions.

1Friction cannot be switched oﬀ in experiments. However, experiments which study universal properties of the jamming transition are also discussed in section 4.1.

15 4. The jamming transition

The jamming transition has been investigated thoroughly and identiﬁed as a second-order, i.e., continuous, phase transition. The extraction of critical exponents by numerical simulations conﬁrms a continuous jamming scenario with a zero-stress critical point, e.g., Olsson and Teitel [2007]; Otsuki and Hayakawa [2008]; Heussinger and Barrat [2009]; Otsuki and Hayakawa [2009]; Otsuki et al. [2010]; Hatano [2010]; Heussinger et al. [2010]; Vågberg et al. [2011].

The macroscopic mechanical properties of a granular assembly are directly linked to the jamming point. At a packing fraction, φJ, a rigidity transition takes place and we expect solid like behavior above φJ, i.e., ﬁnite and positive shear and bulk moduli. Indeed, in an isotropic system, the packing fractions where shear and bulk moduli become nonzero equal φJ [O’Hern et al., 2003]. φJ is considered the critical packing fraction for the onset of rigidity, with a yield stress, ΣY, that vanishes at point J. Below φJ, only ﬂuid states can exist and the stress is determined by the strain rate, γ˙ , and a viscosity, η, that depends on the packing fraction2.

In the vicinity of φJ, both states, solid and ﬂuid, obey scaling laws with distance to φJ. Guided by numerical simulations, critical exponents for the behavior of yield stress, viscosity, and stress are determined [Otsuki and Hayakawa, 2008, 2009; Otsuki et al., 2010; Hatano, 2010]. For a comprehensive review on scaling exponents, see [Dinkgreve et al., 2015]. Here, we state the exponents for soft dry granular particles with inertia in two dimensions. In the solid state above jamming, φ > φJ, a ﬁnite yield stress emerges with distance to jamming:

β ΣY ∝ (φ − φJ) , (4.4)

with an exponent, β = 1. In the ﬂuid state, φ < φJ, the viscosity, η, diverges with the distance to jamming in the limit of zero strain rate:

−α η ∝ (φJ − φ) , (4.5)

with an exponent, α = 4. At the critical packing fraction, φ = φJ, the shear

2In a Newtonian fluid, the viscosity is defined by the relation Σ = ηγ˙ . In a dry granular medium, there is no linear dependence between shear stress and strain rate and the so- called generalized viscosity is defined by Σ = ηγ˙ 2, see section 5.1 for further information.

16 4.1. Frictionless jamming stress depends on the strain rate and scales as:

χ Σ(φJ) ∝ γ˙ , (4.6) with an exponent, χ = 2/5.

The rheological properties close to the jamming transition are related to the particles and their motion. While approaching the jamming transition, particles tend to move cooperatively when the granular medium is sheared: The motion of one particle induces motion of other particles. This cooperative motion is rationalized by a length scale, ξ, which diverges at the jamming transition:

−ν ξ ∝ |φ − φJ| , (4.7) with an exponent, ν, typically estimated as ν ∈ [0.1, 1] and possibly different above and below jamming; For different estimates see, e.g, [Lechenault et al., 2008; Heussinger and Barrat, 2009; Vågberg et al., 2011; Liu et al., 2014; Kawasaki et al., 2015]. The diverging length scale near the transition from arrest to flow is universal. A diverging length scale near jamming is observed for a variety of systems: Flows down an incline3, e.g., [Pouliquen, 2004; Baran et al., 2006; Bon- noit et al., 2010], simple shear geometry, e.g., [Olsson and Teitel, 2007; Goyon et al., 2008; Heussinger and Barrat, 2009; Lemaître and Caroli, 2009; Liu et al., 2014], Poiseuille flow, e.g., [Goyon et al., 2008; Tewari et al., 2009], or a vibrated monolayer of brass cylinders, e.g., [Lechenault et al., 2008].

The existence of a length scale points towards finite size effects. Finite size effects are predicted by theory [Bocquet et al., 2009] and observed in experiments [Goyon et al., 2008]. Systematic studies of finite size effects to gain insights in the jamming transition are done numerically, e.g., [Goodrich et al., 2012; Liu et al., 2014; Vågberg et al., 2014; Goodrich et al., 2014]. In Kawasaki et al. [2015], the authors argue for possible finite size effects when the length scale becomes comparable to the linear extension of the system.

3In this example the packing fraction is not a control parameter. The angle of the incline can control whether ﬂow or arrest of a deposited material is achieved.

17 4. The jamming transition

4.2. Frictional jamming

Friction changes the topology of the jamming phase diagram fundamentally. A theoretical description of frictional hard spheres shows that jamming occurs in a range of φrlp up to φrcp, where φrlp is called random loose packing [Song et al.,

2008]. In two dimensions, φrlp ≈ 0.767 [Silbert, 2010]. The range from φrcp up to φrlp is considerably larger (≈ 1%) than what can be produced for frictionless particles by changes in the protocol, see section 4.1. While φrlp is the lower bound for infinite friction coefficient, φrcp is the upper bound that is approached by φJ in the frictionless limit. That is, the jamming packing fraction is a function of the friction coefficient, φJ = φJ(µ). Thus, friction opens a wide room for jamming to take place. Studies with photoelastic disks under external stress, see [Bi et al., 2011], show that friction allows for anisotropic jammed states: States where stress transmission is anisotropic, which implies the expectation of a rigid response along restricted directions but fluid response in others [Cates et al., 1998]. Anisotropic jammed states are observed at packing fractions lower than φJ, which is the isotropic jamming packing fraction. The anisotropic jammed states are called shear jammed and are observed above a packing fraction φSJ < φJ. At φSJ, the yield stress of the jammed state is finite and vanishes discontinuously upon decreasing the packing fraction. For a schematic phase diagram for the frictional jamming scenario inferred from Bi et al. [2011], see figure 4.3. Shear jamming is also found in corn starch experiments [Fall et al., 2015; Jiang et al., 2015] and rheometry with PMMA spheres confirms the extraordinary impact of frictional contacts [Guy et al., 2015; Pan et al., 2015]. Friction enriches rheological properties close to jamming by additional dynamical phases of transient flow when granular media are sheared [Ciamarra et al., 2011]. In contrast to frictionless systems, frictional systems show clear first- order like phenomena like discontinuous flow curves and hysteresis [Otsuki and

Hayakawa, 2011]. Moreover, the authors showed that φJ loses its exceptional meaning to characterize scaling relations. The scaling exponents are not aﬀected by friction but φJ alone does not suﬃce to characterize the scaling of the divergence of the viscosity and of the yield stress.

18 4.2. Frictional jamming

fluid

Σ shear jammed jammed φ φ φ SJ J rcp φ Figure 4.3.: Schematic jamming phase diagram for frictional particles following Bi et al. [2011]. Anisotropic shear jammed states exist below φJ < φrcp. Jammed states with the lowest packing fraction possible, φSJ, possess a ﬁnite yield stress, ΣY 6= 0.

5. Shear rheology

This Chapter deals with the rheological investigation of dense granular matter. In section 5.1, we explain how rheological data is acquired experimentally. We give an overview of the possible outcomes, which are collected in flow curves that describe the relation between shear stress and strain rate. Complex fluids exhibit different flow regimes, which are distinguished by the relation between stress and strain rate. In sections 5.2 and 5.3, we deal with flow regimes of granular media. A transition between these flow regimes, called shear thickening, is treated in section 5.4. Close to transitions, unsteady flow emerges, which is discussed in section 5.5.

5.1. Rheometry and ﬂow curves — an overview

The mechanical response to strain or stress in the proximity of the jamming transition is probed in rheometers or with computer simulations. The measured response characterizes flow properties and provides insight into the jamming transition itself. A rheometer is sketched in figure 5.1. The probe is confined in a fixed volume by a pair of plates1. Hence, the packing fraction of the particles in the volume is a control parameter. One confining plate remains fixed while the other plate rotates with either constant angular velocity (constant strain rate) or constant torque (constant shear stress). The complementary quantity is measured in terms of the response of the medium, which is characteristic of the flow regime. Typical flow curves of frictionless granular media with different packing fractions are shown in figure 5.2 (left). When the probe is dense, athermal, dry, and composed of soft particles three different flow regimes are identified [Campbell,

1There are also rheometers that guarantee ﬁxed normal load. Then the volume of the probe changes in course of the experiment.

21 5. Shear rheology

Figure 5.1.: Sketch of a rheometer. A ﬁxed volume between plates (closed by dotted lines) is ﬁlled with particles (blue). The lower plate remains at rest and the upper plate rotates with constant angular velocity or torque (indicated by arrow).

2002; Chialvo et al., 2012]:

(i) Inertial or Bagnold ﬂow2 at low strain rate and low packing fraction, φ <

φJ ,

(ii) quasistatic ﬂow at low strain rate and densely packed media, φ > φJ , and

(iii) elastic-inertial ﬂow at large strain rate.

Both, quasistatic and elastic-inertial flow, are referred to as plastic flow. The flow curves can be utilized to characterize the jamming transition. The scaling laws of yield stress and viscosity, which are discussed in section 4.1, allow for a determination of the jamming packing fraction and scaling exponents.

Above φJ, in the quasistatic regime, see 5.2 (left, ii), the stress plateau at low strain rate equals the yield stress. The yield stress vanishes at φJ and grows 2 above. Figure 5.2 (right) shows the generalized viscosity, η = Σ/γ˙ . Below φJ, in

2The interaction and dissipation mechanisms (e.g., drag forces, friction, etc.) influence the rheological response and determine the flow curve [Vågberg et al., 2014]. As a consequence, e.g., when the grains are suspended in a Newtonian fluid, a Newtonian regime, i.e., Σ ∝ γ˙ , appears instead of Bagnold flow.

22 5.1. Rheometry and ﬂow curves — an overview

1010 ·0 (iii) -2 ∝ γ 10 ·1/2 ∝ γ 8 (ii) 10 ϕ -4 10 106 2 · γ / Σ Σ =

(i) η 104 10-6 ϕ ·2 ∝ γ 102 10-8

100 -6 -4 -2 10-6 10-4 10-2 10 10 10 · γ· γ Figure 5.2.: Left: Schematic flow curves according to the model in Grob et al. [2014], mimicking frictionless particle flow. The arrow indicates increasing packing fraction and φJ is marked by a thick line. Three flow regimes are measured: (i) inertial flow, (ii) quasistatic flow, and (iii) elastic-inertial flow. Right: Generalized viscosity corresponding to the flow curves on the left. Units set by the particle properties according to [Grob et al., 2014].

23 5. Shear rheology the inertial regime, the generalized viscosity plateaus at a value that depends on the packing fraction. As φJ is approached, the height of the plateau diverges in the limit of zero strain rate. Both, viscosity and yield stress, allow for estimates of the jamming packing fraction φJ. The interplay of time scales set by the experimental setup, i.e., strain rate or shear stress, and particle stiﬀness determine the rheology. The elastic properties of the constituents tune the ﬂow characteristics and crossovers between the regimes [Campbell, 2002].

5.2. Inertial ﬂow

Stiff grains or slow deformation imply that the time scale set by the stiffness of √ the particles, k−1, is smaller than the time scale set by the strain rate, γ˙ −1. When the only relevant time scale is set by the strain rate, i.e., the deformation can be considered as sufficiently slow or the grains as sufficiently hard, stress scales as [Bagnold, 1954] Σ ∝ γ˙ 2. (5.1)

Inertial flow can be thought of as ballistic motion of grains interrupted by collisions. It is this ballistic motion that governs the momentum transport in the inertial flow regime [Bagnold, 1954; Campbell, 2011]. Numerical simulations of soft spheres show that, in the inertial flow regime, the duration of collisions approaches the expected binary collision time set by the model parameters of the particles [Campbell, 2002]. Therefore, collisions are mainly binary. The binary character of collisions is a necessary precondition for the applicability of the kinetic theory of granular gases as a description of flow, see Brilliantov and Pöschel [2010]. In the kinetic theory, particles are modeled by infinitely stiff spheres, i.e., hard spheres, which collide with a collision rule and a coefficient of restitution. Hard spheres do not overlap and collisions are instantaneous. In practice, the particle stiffness is finite and therefore the hard sphere limit cannot be reached. A measure to characterize the ratio of inertial forces — set by the strain rate — to confining forces — set by the pressure in the system, P — is the dimensionless

24 5.3. Plastic ﬂow inertial number [Midi, 2004; Da Cruz, 2004; Da Cruz et al., 2005]:

q I =γ ˙ m/P , (5.2) with particles of mass m. In contrast to the kinetic theory of granular gases, this rheological approach is a route of modeling flow that incorporates the finite particle stiffness and does not rely on a collision mechanism. In a hard particle system, inertia dominates since interparticle forces are absent in instantaneous collisions, i.e., the inertial number is large. In a system of soft particles, this is not necessarily true and confining forces can dominate. Therefore, the inertial number is used to describe several flow regimes, not only inertial flow. The inertial number proved to be a useful tool for the description of constitutive equations and, in particular, of friction laws that relate macroscopic shear stress to pressure, e.g., µ = Σ/P [Midi, 2004; Da Cruz et al., 2005; Chialvo et al., 2012]. The friction laws are known as µ(I)-rheology and used in hydrodynamic descriptions of granular flow, see, e.g., [Jop et al., 2006]. However, for fluid mechanics applications, the simple µ(I)-rheology is of limited applicability since it is mathematically ill-posed for large and low inertial numbers [Barker et al., 2015]. Non-local rheology, which can be regarded as series expansions where heterogeneities are taken into account, extends the idea of µ(I)-rheology [Volfson et al., 2003; Aranson et al., 2008; Bouzid et al., 2013, 2015].

5.3. Plastic ﬂow

Plastic flow is subdivided into quasistatic flow and elastic-inertial flow. In the quasistatic flow regime, the strain rate is low and only the time scale set by the stiffness of the particles is relevant. The particle stiffness and packing fraction determine the stress plateau in the flow curve. The value of the stress plateau equals the yield stress:

Σ = ΣY. (5.3)

In the elastic-inertial ﬂow, the shear stress depends on the strain rate as a square root, see, e.g., [Bocquet et al., 2009; Lemaître and Caroli, 2009; Olsson and Teitel,

25 5. Shear rheology

2012; Chialvo et al., 2012], as shown in ﬁgure 5.2 (left, iii):

Σ ∝ γ˙ 1/2. (5.4)

In the plastic ﬂow regime, the particles’ elastic properties are relevant and interparticle contacts are transmitters of stress and momentum [Campbell, 2006]. The contact duration exceeds the binary collision time by up to two orders of magnitude and contacts are not binary [Campbell, 2002]. This requires ﬂow models other than the kinetic theory of granular gases and µ(I)-rheology.

Early work on the plastic deformation of metallic glasses led to the picture that irreversible plastic events release stress in localized regions of size of a few particles — in so-called shear transformation zones [Argon, 1979]. The irreversible yielding of a shear transformation zone is called shear transformation. The idea of shear transformations was developed further to describe plastic flow and macroscopic yielding. Plastic flow is considered as a succession of elastic deformations, which accumulate potential energy, until shear transformations release energy. Shear transformations trigger each other in a cascade [Maloney and Lemaître, 2006]. When average characteristics of shear transformations are incorporated in a description of flow, even macroscopic yielding of a jammed medium upon stress increase can be explained [Falk and Langer, 1997]. A similar description is used to describe the dynamics of locally yielding zones in generic soft glassy matter [Sollich et al., 1997]. The authors point out that many soft materials show structural disorder and metastability, which lead to the glass like behavior when deformed. A characteristic length for cooperativity larger than the size of the molecules was evidenced in experiments [Goyon et al., 2008]. In line with this insight, a nonlocal constitutive law with a local rate of plastic events has been derived [Bocquet et al., 2009]. The non-locality is expressed by a flow cooperativity length that diverges in the quasistatic limit of zero strain rate. This implies finite size effects of flow and a dynamic yield stress at a critical point for a second-order phase transition. Elastic deformations interrupted by plastic rearrangements and a growing correlation length have been reported in several studies of dense flows, e.g., [Heussinger and Barrat, 2009].

26 5.4. Shear thickening

10-2

10-3

10-4

(arb. units) ·2 Σ ∝γ 10-5

10-5 10-4 10-3 10-2 γ· (arb. units) Figure 5.3.: Flow curves of a shear thickening ﬂuid display continuous shear thickening (green) and discontinuous shear thickening (brown). The arrows indicate possible hysteresis in the discontinuous scenario.

5.4. Shear thickening

Flow regimes, e.g., inertial and plastic flow, differ fundamentally from each other with respect to transport of particles, momentum, and stress. The transitions between flow regimes are highly debated since the triggering mechanisms can be manifold. Shear thickening is such a transition and describes the increase of viscosity with shear stress or strain rate. Figure 5.3 shows potential flow curves of a shear thickening fluid. The scenario in figure 5.3 (green thick line) with a flow curve with finite, positive, and excessive increase (in this logarithmic representation a slope larger than 2), is termed continuous shear thickening. When the flow curve exhibits parts with finite jumps shear thickening is discontinuous, see figure 5.3 (brown line). In the discontinuous scenario, the continuous increase or decrease of the strain rate leads to jumps from one flow branch to the other, as indicated by the arrows in figure 5.3. The shear stress, and thereby the viscosity, changes discontinuously. The phenomenon of shear thickening is absent in the

27 5. Shear rheology

flow curves of frictionless granular matter as illustrated in figure 5.2. Approaches that try to explain shear thickening of a complex fluid and the shape of a flow curve, as shown in figure 5.3, stem from a broad variety of studies. In the following, we discuss studies that exhibit shear thickening but differ by the (experimentally, numerically, or analytically) examined system. Experi- ments are mainly conducted on suspensions, i.e., heterogeneous mixtures of solid particles floating in a solvent. Examples are an aqueous solution with suspended photoelastic disks or micro meter sized silica spheres. These examples possess an important difference: large photoelastic disks are granular grains and athermal contrasting small silica spheres that are only of micro meter size and experience Brownian motion [Brown, 1828]. Solid Brownian particles, e.g., micro meter sized silica spheres or micelles suspended in a solvent, are called colloids. A general and abstract argument for discontinuous shear thickening flow curves is the coexistence of differently flowing phases within the system. In this case, local flow curves for each phase add up to the macroscopic flow curve that can be described by a so-called s-shaped or sigmoidal flow curve [Olmsted, 1999]. In particular, shear localization leads to non-monotonic flow curves [Olmsted, 2008; Schall and van Hecke, 2010]. Indeed, localized phases are observed in experiments and are accompanied with shear thickening. In micellar solution, shear thickening is associated with localization of an anisotropic phase as an evidence of coexisting phases [Berret et al., 2002]. In dense suspensions, shear thickening and shear jamming are reported, but both pinned to shear localization [Fall et al., 2008]. The authors argue that shear jamming results from dilatancy and confinement, i.e., shear thickening turns into shear jamming because the system’s finite size. In this paragraph, we treat arguments that rely on the presence of a suspension. An explanation for shear thickening suspensions are hydrodynamic interactions [Wagner and Brady, 2009]. Dynamically correlated clusters, which enhance the particles non-affine motion and thereby enhance dissipation, are another mechanism [Andreotti et al., 2012; Heussinger, 2013]. In suspensions, where the inertia of colloids is not negligible, shear thickening is argued to be due to anisotropy in the microstructure which creates an effectively larger packing fraction [Picano et al., 2013]. A conclusive phase diagram, including a shear thickening regime, for Newtonian suspensions with friction and inertia was proposed recently but does not give an explanation for flows of dry granular media [Ness and Sun, 2015].

28 5.4. Shear thickening

Also, the contact network between the particles is an important mediator for shear thickening. In this paragraph, we deal with the importance of contacts. The role of frictional contacts is highlighted by rheometry with PMMA spheres, numerical simulations [Guy et al., 2015; Ness and Sun, 2016], and with colloidal silica particles [Royer et al., 2016]. Shear thickening is explained by different mechanisms of particle contacts [Wyart and Cates, 2014]: When particles are pushed together with large enough forces, frictional contacts become relevant and support larger load than lubricated contacts of particles which are just slightly pushed together. However, it remains elusive which kind of shear thickening, i.e., continuous or discontinuous, is realized in an experiment or simulation and by which mechanism it is implied. Contact forces dominate continuous shear thickening in suspension, as the mildest3 form of shear thickening [Lin et al., 2015]. In Fernandez et al. [2013]; Mari et al. [2014], both, continuous and discontinuous shear thickening, were identified to occur based on frictional contacts and lubrication forces but controlled by packing fraction, i.e., shear thickening becomes stronger when the packing fraction increases and eventually turns from continuous to discontinuous. The packing fraction controls the number of contacts and thus the packing fraction is a crucial control parameter for shear thickening media, see also Brown and Jaeger [2009]; Seto et al. [2013]. The discontinuous scenario is accompanied with phenomenology reminiscent of first-order phase transitions in equilibrium statistical mechanics, e.g., hysteresis in simulations of dry granular media [Otsuki and Hayakawa, 2011]. Moreover, the interaction mechanisms that is minimally required for shear thickening is controversial. In Wyart and Cates [2014], the authors argued that a microscopic stress scale, which distinguishes lubricated from frictional contacts, is necessary for shear thickening. In contrast, in Otsuki and Hayakawa [2011], the authors do not dwell on such a mechanisms but use the same simplified approach as presented in Chapter 3. As the discussion points out, shear thickening is observed in different experimental, numerical, and theoretical investigations. A host of different mechanisms are found to be the origin for shear thickening, e.g., friction, lubrication forces, correlated cluster, shear localization, and other heterogeneities. Heterogeneities lead to gradients and result in unsteady flow. Heterogeneities and unsteady flow have not been discussed in dry granular media. Also, many of the arguments

3The viscosity grows less in continuous shear thickening than in the discontinuous shear thickening.

29 5. Shear rheology above do not hold when the suspending ﬂuid has zero or negligible viscosity.

5.5. Unsteady ﬂow

The sections on inertial flow 5.2 and on plastic flow 5.3 deal with flow states that do not show time dependent behavior — except for fluctuations. Due to the fact that the flowing medium is composed of grains, discrete interactions, i.e., collisions, make fluctuations an inherent feature of the dynamics and thus will not be considered as a signature of unsteady flow. The flow is steady when the system is tuned in the inertial or plastic flow regime. In the proximity of transitions between inertial or plastic flow, unsteady behavior can emerge. Three possible scenarios for unsteady flow are presented in the following:

(i) the coexistence of states, which are realized alternately as time progresses,

(ii) oscillations with a systematic pattern, and

(iii) chaotic response and irregular dynamics.

A rigorous characterization of these different scenarios is far from trivial and thus a distinction between these states is a delicate task and not always unambiguous. Figure 5.4 shows time series of the different unsteady flow scenarios. Coexistence is given when metastable states are realized alternately, e.g., in the course of an experiment or a simulation, two differently flowing phases alternate, see figure 5.4 (i). In numerical studies of frictional particles, metastable flow states are probed and successive alternations between plastic and inertial flow are observed near a critical packing fraction [Aharonov and Sparks, 1999; Mari et al., 2014]. Similar features are found in three dimensional simulations, where the evolution of contact number and pressure are shown to be linked; both fluctuate the most at a critical packing fraction [Chialvo et al., 2012]. Metastable states and hysteresis due to shear localization are found experimentally, e.g., [Chen et al., 1992; Berret and Porte, 1999]. In a shear thickening regime, the stress distribution shows two peaks, which is interpreted as coexistence of differently flowing states [Heussinger, 2013]. Therefore, the shear thickening regime presented by the author is only apparent as it is constituted not by persistent, but alternating flow states.

30 5.5. Unsteady ﬂow

10-3 (i)

Σ 10-4

10-5 10-3 10 20 30 40 (ii)

Σ 10-4

10-5 10-3 16 17 18 19 (iii)

Σ 10-4

10-5 5 6 7 8 9 10 γ

Figure 5.4.: Different scenarios for unsteady flow: (i) coexistence, (ii) flow with systematic patterns, and (iii) chaotic response. Units set by the particle properties according to [Grob et al., 2014]. Simulation parameters: φ = 0.7975 and (i) N = 8000, imposed strain rate γ˙ = 1.1 × 10−4, (ii) N = 80000, imposed strain rate γ˙ = 1.25 × 10−4, and (iii) N = 80000, imposed shear stress Σ = 6.28 × 10−5.

31 5. Shear rheology

Oscillations with a systematic pattern are another realization of unsteady flow, see figure 5.4 (ii). A discussion on sigmoidal flow curves, their relation to frictional contacts and the potential of shear instabilities leading to oscillations is given in Bashkirtseva et al. [2009]. The authors also present a phenomenological model for the flow of a complex fluid, which, locally, shows a nonlinear viscoelastic response to stress — ultimately leading to global oscillations. Colloidal experiments show non-Gaussian stress fluctuations with heavy tailed distributions when the system is driven in an apparent shear thickening regime [Lootens et al., 2003]. Experiments on micellar solutions evidence shear thickening, hysteresis and oscillating behavior [Fischer, 2000; Fernández et al., 2009; Lutz-Bueno et al., 2013]. The authors conclude that the system does not settle into a steady state because different phases with different flow characteristics exist. The emergence and de- struction of these phases are dynamically induced by shear in an unstable flow regime. Time dependent flow is found near jamming of gravity driven flow down an incline and shows temporally heterogeneous dynamics and intermittent behavior characterized by spatial heterogeneities visible in the contact network [Silbert, 2005]. A simple rheological model predicts oscillatory flow and a sigmoidal flow curve [Head et al., 2001, 2002]. The scenario of flow with chaotic dynamics, see figure 5.4 (iii), appears to be the least documented. A simple scalar model of a shear thickening material predicts rheological chaos as a result of flow instabilities [Cates et al., 2002]. Also, fluid dynamic models show complex rheological behavior in shear thickening states [Hess et al., 2006]. In corn starch experiments, unpredictable rheological response is measured where discontinuous shear thickening is expected [Hermes et al., 2015].

32 6. Active microrheology

Microrheology examines rheological properties in complex fluids by the dynamics of a suspended probe particle. A comprehensive review is given by Squires and Mason [2010]. In so-called passive microrheology, the motion of the probe particle is induced by the temperature of the suspending fluid, T . The simplest microrhe- ological equation is the Stokes-Einstein relation, equation 6.1, which relates the viscosity of a Newtonian fluid in equilibrium, η, to the diffusion constant, D:

k T D = B , (6.1) 6πrη with the Boltzmann constant, kB, and the radius of the probe particle, r [Einstein, 1905; Sutherland, 1905]. The mean square displacement expressed in terms of the diffusion constant, h∆x2(t)i = 2Dt, is the link between the position of the probe particle as a microscopic quantity and the viscosity as a rheological quantity. Thereby, the Newtonian viscosity can be inferred from the probe particle’s fluctuations. Seminal work by Mason and Weitz [1995] examines the relation of the mean square displacement and the frequency dependent complex shear modulus, G?(ω)1, of a non-Newtonian fluid, which describes the linear rheological relation between shear stress and shear strain, completely [Mason and Weitz, 1995]. When external driving is applied to the probe particle microrheology is called active. The active forcing allows to go beyond the linear-response regime and to examine linear and nonlinear rheological response. Nonlinear phenomena are, e.g., shear thinning, shear thickening, and the granular glass transition, a rigidity transition of a stochastically driven granular fluid [Kranz et al., 2010; Sperl et al., 2012; Kranz et al., 2013]. To study the granular rigidity transition, Can- delier and Dauchot [2009, 2010] conducted experiments on a horizontal layer of vibrated granular particles and examined the dynamics of an externally driven

1Real and imaginary part of G?(ω) correspond to the elastic and viscous modulus, respectively.

33 6. Active microrheology probe particle. Two qualitatively diﬀerent regimes, seperated by a packing fraction dependent force threshold, which diverges at the granular glass transition, are reported: a ﬂuidized regime with continuous motion and a solid-like regime with intermittend motion. In general, the external driving is realized with a force, F , or a velocity, v, which is imposed on the probe particle and the complementary quantity is measured. This gives access to the mobility, ν, via the velocity-force relation

v = νF. (6.2)

A velocity-force relation is in general nonlinear due to force thickening, i.e., a sublinear velocity-force relation, or force thinning, i.e., a superlinear velocity-force relation. In this context, the inverse of the mobility is called friction coeﬃcient, ζ2. The frequency dependent generalized Stokes mobility, ν?(ω), relates the mobility to the complex viscosity, η?(ω) = −iω−1G?(ω):

η ν?(ω) = ν, (6.3) η?(ω) with the frequency, ω, which is set by the velocity of the probe particle. In shear rheology, as discussed in Chapter 5, the strain rate is homogeneous (in a stationary state in simple shear geometry). In microrheology, however, shear is applied locally, i.e., the medium near the probe particle can be far out of equilibrium while further away from the probe particle, the medium can still remain undeformed or unstressed. Therefore, a quantitative comparison between shear rheology and microrheology is, in general, not appropriate. Nevertheless, microrheology gives insight into microstructural properties in force thinning regimes [Sriram et al., 2009] or growing length scales close to jamming (or the granular glass transition) [Candelier and Dauchot, 2009, 2010]. The review Reichhardt and Reichhardt [2014] discusses what can be learned about jamming with the help of externally driven probes and Puertas and Voigtmann [2014] review microrheology on colloids and discuss granular systems, too.

2The friction coeﬃcient is not to be confused with Coulomb’s friction parameter, which is called µ.

34 Part II.

Results

This part contains the results of this thesis, which are presented as separate articles. Chapter 7 contains the main achievement of this thesis: studies on sheared frictional granular media. In section 7.1, we present the article “Jamming of frictional particles: A nonequilibium first-order phase transition”, [Grob et al., 2014], which discusses the jamming transition of sheared frictional granular particles in small simulation cells. The article in section 7.2, “Rheological chaos of frictional grains”, [Grob et al., 2016], treats unsteady flow near a shear induced jamming transition in large simulation cells. We present numerical findings that are in accordance with the prediction of a simple model that is developed in the same study. The stability analysis of the solutions of the model is presented in appendix A. In section 7.3, we present the manuscript “Unsteady rheology and heterogeneous flow of dry frictional grains”, a detailed study, which links the preceding studies together and provides a detailed description of heterogeneities and time dependent flow close to jamming. Chapter 8 contains results of a study that was carried out as a side project during the period of this thesis. The numerical foundations of this study, which was developed before the period of this thesis, is described in [Fiege et al., 2012]. We use an event-driven simulation scheme that accounts for a resistive drag force3 and introduces an actively driven probe particle of which we investigate the nonlinear velocity-force relation close to the granular glass transition. The numerical results are related to analytical results by Wang and Sperl in the article “Active microrheology of driven granular particles” [Wang et al., 2014].

3 When a particle moves ballistically with velocity, v, a drag force, FD, is a resistive force proportional to the particle’s velocity: FD ∝ −v; it is motivated by, e.g., a ﬂuid that surrounds the particles or, in a dry system, friction with a bottom plate on which the experiment is built.

7. Shear rheology of frictional grains

7.1. Jamming of frictional particles: A nonequilibium ﬁrst-order phase transition

Reprinted article with permission from Grob, Matthias and Heussinger, Claus and Zippelius, Annette Physical Review E 89 050201 (2014) http://dx.doi.org/10.1103/PhysRevE.89.050201 Copyright (2014) by the American Physical Society.

RAPID COMMUNICATIONS

PHYSICAL REVIEW E 89, 050201(R) (2014)

Jamming of frictional particles: A nonequilibrium ﬁrst-order phase transition

Matthias Grob,1 Claus Heussinger,2 and Annette Zippelius1,2 1Max Planck Institute for Dynamics and Self-Organization, Am Faßberg 17, 37073 Gottingen,¨ Germany 2Institute for Theoretical Physics, Georg-August University of Gottingen,¨ Friedrich-Hund Platz 1, 37077 Gottingen,¨ Germany (Received 21 November 2013; revised manuscript received 12 February 2014; published 27 May 2014) We propose a phase diagram for the shear flow of dry granular particles in two dimensions based on simulations and a phenomenological Landau theory for a nonequilibrium first-order phase transition. Our approach incorporates both frictional as well as frictionless particles. The most important feature of the frictional phase diagram is reentrant flow and a critical jamming point at finite stress. In the frictionless limit the regime of reentrance vanishes and the jamming transition is continuous with a critical point at zero stress. The jamming phase diagrams derived from the model agree with the experiments of Bi et al. [Nature (London) 480, 355 (2011)] and brings together previously conflicting numerical results.

DOI: 10.1103/PhysRevE.89.050201 PACS number(s): 45.70.−n, 66.20.Cy, 83.60.Rs

Random close packing is the point at which hard particles all have the same mass m = 1, but are polydisperse = spherical—and frictionless—particles generally jam into a sta- in size: 2000 particles each for diameter d 0.7,0.8,0.9,1.0. = N 2 2 ble heap. It is now known that the precise close-packing density The particle volume fraction is defined as ϕ i=1 πdi /4L . (n) (t) ϕrcp depends on the preparation protocol [1]. Nevertheless, this Normal and tangential forces, f and f , are modeled with variability is small when compared to frictional systems, i.e., linear springs of unit strength for both elastic as well as viscous systems where particles not only transmit normal forces but contributions. (Thereby units of time, length, and mass have also tangential forces among themselves. Indeed, frictional been fixed.) Coulomb friction is implemented with friction systems can jam at densities anywhere between random-close, parameter μ = 2[12]. In the strain-controlled simulations, we random-loose, or even random-very-loose packing [2,3]. In prepare the system with a velocity profile vflow = γ˙(0)yeˆx this Rapid Communication we deal with the flow properties initially. Subsequently the shear rate is implemented with of frictional granular systems, where the jamming transition Lees-Edwards boundary conditions [13] until a total strain can be studied by monitoring the flow curves, i.e., the stress- of 200% is achieved after time T . Whenever the strain rate is strain rate relations σ(˙γ ). Previous simulations performed in changed to a new value, we wait for a time ∼0.5T to allow for the hard-particle limit [4,5] do not observe any qualitative the decay of transients. In the stress-controlled simulations, a difference between frictionless and frictional systems, other boundary layer of particles is frozen and the boundary at the than a mere shift of the critical density from ϕrcp to ϕJ (μ), top is moved with a force σLeˆx , whereas the bottom plate which depends on the friction coefficient μ of the particles. remains at rest. Similar results, accounting for particle stiffness, are presented In the strain-controlled simulations we impose the strain in Refs. [6,7]. Quite in contrast, Otsuki et al. [8] recently rateγ ˙ and measure the response, the shear stress σ(˙γ ), for observed a discontinuous jump in the flow curves of the a range of packing fractions 0.78 ϕ 0.82. Thereby the frictional system, which is absent in the frictionless analog [9]. system is forced to flow for all packing fractions; the resulting In addition, they find not one but three characteristic densities flow curves are shown in Fig. 1. for the jamming transition, which degenerate into random close We observe three different regimes. For low packing packing when μ → 0. Similarly, Ciamarra et al. [10] observe fraction, the system shows a smooth crossover from Bagnold three (but different) jamming transitions. Experimentally, Bi scaling, σ = ηγ˙ 2 (called “inertial flow”) to σ ∝ γ˙ 1/2 (called et al. [11] present a jamming phase diagram with a nontrivial “plastic flow”). As the packing fraction is increased, we (reentrance) topology that is not present in the frictionless observe a transition to hysteretic behavior [8]: Decreasing the scenario. strain rate from high values, the system jumps discontinuously These latter results hint at friction being a nontrivial to the lower branch. Similarly, increasing the strain rate from and indeed “relevant” perturbation to the jamming behavior low values, a jump to the upper branch is observed. A well of granular particles. Unfortunately, several inconsistencies developed hysteresis loop is shown in the inset of Fig. 1.The remain unresolved. For example, the phase diagram in [10] onset of hysteresis defines the critical density ϕc. We estimate is different from [11] and does not show stress jumps as its value ϕc between 0.7925 and 0.795 by visual inspection observed in [8]. This points towards a more fundamental of the flow curves as described in the Supplemental Material lack of understanding of the specific role of friction in [14]. As ϕ is increased beyond the critical value ϕc,thejump these systems. What is the difference between frictional and to the lower branch happens at smaller and smallerγ ˙, until at frictionless jamming? By combining mathematical modeling ϕσ , the upper branch first extends to zero strain rate, implying with strain- and stress-controlled simulations we propose a the existence of a yield stress, σyield.Forϕc <ϕ<ϕσ ,the jamming scenario that not only encompasses frictional as well strain rate for the jump to the lower branch,γ ˙σ ∝ ϕσ − ϕ, as frictionless systems, but also allows one to bring together scales linearly with the distance to ϕ which allows us ∼ σ previously conflicting results. to determine ϕσ = 0.8003. Finally at ϕη, the generalized = 2 We simulate a two-dimensional system of N 8000 soft, viscosity η = σ/γ˙ diverges and for ϕ>ϕη only plastic flow −4 frictional particles in a square box of linear dimension L.The is observed. The scaling of the viscosity η ∝ (ϕη − ϕ) is

GROB, HEUSSINGER, AND ZIPPELIUS PHYSICAL REVIEW E 89, 050201(R) (2014)

~ γ·1/2 -2 10 10-2

-4 10 10-4 σ 10-3 σ 10-6 10-6 10-5 ·2 b > 0 b = 0 ~ γ 10-7 -8 10 -8 10-6 10-5 10-4 10-3 10

-5 -4 -3 -2 -1 10 10 10 10 10 -6 -4 -2 -6 -4 -2 γ· 10 10 10 10 10 10 γ· γ· FIG. 1. (Color online) Flow curves σ (˙γ ) for different packing fractions ϕ = 0.78,0.7925,0.795,0.7975,0.79875,0.80,0.82 (from FIG. 2. (Color online) Flow curves of the simple model, Eq. (1). bottom to top). Main part: Flow curves obtained by decreasingγ ˙. Left: Frictional scenario with range of packing fractions as in Fig. 1; Inset: Example of a hysteresis loop for ϕ = 0.80. φc is indicated by the red line, φσ by the blue line, and φη by the green line. Right: Flow curves for frictionless particles of the simple model = = = = ∼ implemented with b 0andϕc ϕσ ϕη 0.8433; critical flow in agreement with previous results [8] and yields φη = 0.819. curve in red. Note that all three packing fractions are well separated, and furthermore, ϕ is still well below the frictionless jamming η ∼ density at random close packing ϕrcp = 0.8433. The scaling This happens at a density ϕ determined implicitly by a(ϕ ) = plotsγ ˙σ and η are shown in the Supplemental Material [14]. √ σ c = 2/3 All the observations can be explained in the framework of a(ϕσ ) 2 and the yield stress is given by σyield [a/(2c)] . a simple model, which can be viewed as a phenomenological The flow curves can be fitted better, if we allow for weakly Landau theory, that interpolates smoothly between the inertial density dependent coefficients b and c. However, we refrain and the plastic flow regime: from such a fit, because even in its simplest form the model can account for all observed features qualitatively: a critical point = 1/2 − + 2 γ˙(σ) aσ bσ cσ , (1) at ϕc, the appearance of a yield stress at ϕσ , and the divergence of the viscosity at ϕ , ordered such that ϕ <ϕ <ϕ .The a,b,c η c σ η where are coefficients which in general depend on the flow curves for these three packing fractions are highlighted packing fraction. Equation (1) can be taken to result from in Fig. 2 and further illustrated in the Supplemental Material a class of constitutive models that combine hydrodynamic [14]. conservation laws with a microstructural evolution equation The limiting case of frictionless particles can be reached by [15], or from mode-coupling approaches [16]. letting μ → 0. Simulations indicate that in this limit hysteretic The numerical data suggest that the plastic flow regime effects vanish [8,9] and the jamming density is increased is only weakly density dependent for packing fractions approaching random close packing. Within the model this c ϕ considered here, so we take to be independent of for transition can be understood in terms of the variation of two simplicity. In the inertial flow regime, on the other hand, parameters: First b(μ) → 0inEq.(1) implies that the three ϕ we expect to see a divergence of the shear viscosity at η, densities (ϕ ,ϕ ,ϕ ) coincide and second ϕ (μ) → ϕ =∼ a ϕ c σ η η rcp implying that the coefficient of our model vanishes at η and 0.8433. While a μ-dependent ϕ simply shifts the phase a = a ϕ = a |ϕ − ϕ| ϕ − ϕ η changes sign, ( ) 0 η ( η ). The coefficient diagram towards higher densities, the parameter b accounts b is assumed to be at most weakly density dependent. for the more important changes of the topology of the phase The simple model predicts a discontinuous phase transition diagram. The flow curves in this limit are presented in with a critical point in analogy to the van der Waals theory of Fig. 2(right). They present a continuous jamming scenario the liquid-gas transition [see Fig. 2(left)]. The critical point is consistent with previous simulations in inertial [9]aswellas determined by locating a vertical inflection point in the flow overdamped systems [17,18]. ∂ γ = curve. In other words we require σ ˙ 0 and simultaneously What happens in the unstable region? Naively one might ∂ γ = σσ ˙ 0. These two equations together with the constitutive expect “coexistence” of the inertial and the plastic flow regime, equation (1) determine the critical point: b = 3 a(ϕ )2/3c1/3 c 2 c i.e., shear banding. However, this would have to happen = 3 a4/3 with the critical strain rate given byγ ˙c 16 c1/3 and the critical along the vorticity direction [15,19], which is absent in our = 1 a 2/3 stress σc 4 ( c ) .Forϕ>ϕc, the model predicts an unstable two-dimensional setting. Alternative possibilities range from region, where ∂σ γ<˙ 0. This is where the stress jump occurs oscillating to chaotic solutions [20,21]. We will see that, in the simulations. The flow curves of the model are presented instead, the system stops flowing and jams at intermediate in Fig. 2(left), assuming b ≡ bc, and fitting the two constants stress levels. Interestingly, this implies reentrance in the (σ,ϕ) c,a0 to the data. The model predicts a yield stress to first occur, plane with a flowing state both for large and small stress, and when two (positive) zeros for the functionγ ˙(σ) = 0 coincide. a jammed state in between.

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4· 1.5 4· 4· 40000 10 γσ (t) 10 γσ (t) 3.0 10 γσ (t) 1.5 1 2 3 1.0 35000 2.0 10-3 1.0 30000 0.5 1.0 25000 0.5 0 0

σ 20000 0 2×104 4×104 0 2×104 4×104 0 2×104 4×104 15000 -4 4· 12.0 4· 10 10 γσ (t) 10 γσ (t) 10000 4.0 4 5 8.0 T

jam 5000 τ 2.0 4.0 0 0.7925 0.795 0.7975 0.80 0.80125 σ σ σ σ σ 0 1 2 3 4 5 4 4 4 4 ϕ 0 2×10 4×10 0 2×10 4×10 shear stress σ FIG. 4. (Color online) Numerical results for the phase diagram. FIG. 3. (Color online) Time series of the strain rate with packing The mean flow time τ is encoded with color. The flow phase is fraction ϕ = 0.7975 and different, but fixed, stresses σ <σ <σ < jam 1 2 3 indicated in yellow (bright) and the jamming phase in blue (dark). σ <σ (stress values are indicated in Fig. 6). Lower right corner: 4 5 Lines are contours of constant τ . Schematic picture of the jamming time being cut off at the simulation jam time T . within the (thick) red curve, Eq. (1) has no solution: the system jams. Outside the (thin) blue curve, the solution is To address the unstable regime in more detail, we have unique corresponding to either inertial flow (low stress) or performed stress-controlled simulations: The shear stress is plastic flow (high stress). In between, in the shaded region, imposed and we measure the strain rate as a function of the equations allow for two solutions and hence metastable time. The initial configurations are chosen with a flow profile states. Jamming from these metastable states is discontinuous, corresponding to the largest strain rate in the inertial flow i.e., the strain rate jumps to zero from a finite value. At a regime, which was observed previously in the strain-controlled packing fraction ϕσ , a yield stress first appears and grows as ϕ simulations. For a fixed ϕ, several time series are shown in is increased further, giving rise to a kink in the red curve and Fig. 3, representing the different regimes. The lowest value a continuous jamming scenario. Beyond ϕη inertial flow is no → of σ1 is chosen in the inertial flow regime, so that the system longer possible [22]. In the frictionless limit b 0 all these continues to flow for large times. Similarly σ5 is chosen in the different packing fractions merge with ϕrcp giving the phase plastic flow regime and the system continues to flow as well. diagram the simple structure well known from previous work The intermediate value σ3 is chosen in the unstable region and [23] and shown in the inset in Fig. 5. the system immediately jams. Between the jamming and the The presence of long transients is fully consistent with flow regime we find intermediate phases with transient flow the results of Ref. [10]. Due to a restricted stress range in that ultimately stops [10]. those simulations, however, only the upper part of the phase To quantify the different flow regimes, we introduce the diagram is captured and the reentrance behavior is missed. time τjam the system needs to jam. Schematically we expect the To get a better understanding of these transients (or possibly result shown in the lower right corner of Fig. 3: In the jammed phase τjam = 0, whereas in the flow phase τjam is infinite. In between τjam is finite implying transient flow before the system plastic flow jams. We expect τjam to go to zero as the jammed phase is approached and to diverge as the flow phases are approached. metastable Given that the simulation is run for a finite time, the divergence plastic flow should be cut off at the time of the simulation run, T , indicated by the horizontal (red) line in the schematic in the lower right flow corner of Fig. 3. σ jamming These expectations are borne out by the simulations: In jamming Fig. 4 we show a contour plot of τ as a function of ϕ and jam metastable ϕ σ . In the dark blue region, τjam is very small, corresponding to inertial flow j the jammed state. In the bright yellow region τjam exceeds the inertial simulation time; hence this region is identified with the flow flow 0 regime—inertial flow for small σ and plastic flow for large ϕ ϕ ϕ ϕ c σ η rcp σ . The intermediate (red) part of the figure corresponds to the ϕ transient flow regimes. In our simple model, Eq. (1), the jammed state has to FIG. 5. (Color online) Phase diagram of the model (schematic), be identified with the unstable region. It seems furthermore revealing reentrant flow for small and large σ , as well as flow and suggestive to identify the transient flow regime with metastable jam states in the “metastable” regions for frictional particles (main regions. The phase diagram, as predicted by the simple model panel) and the known jamming phase diagram for frictionless particles (with finite b) is shown in Fig. 5 (schematic). In the region (inset).

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a non monotonic relationγ ˙(σ), which can only be observed σ 5 as transient behavior, before the system has settled into a σ stationary state. -3 4 10 σ In conclusion, the goal of this Rapid Communication is 3 to understand the role of friction in the jamming behavior of dry granular matter. To this end we present a theoretical σ 2 model (supplemented by molecular dynamics simulations) σ σ that can reproduce all the phenomenology of simulated flow 1 10-4 ϕ=0.7925 curves (Fig. 2) both for the fully frictional system as well as ϕ=0.795 for the limiting case of frictionless particles. The jamming ϕ=0.79625 phase diagrams derived from the model agree with recent ϕ=0.79875 experiments [11]. The key result is that the transition between ϕ=0.8003 the two jamming scenarios, frictionless and continuous, and -5 10 frictional and discontinuous, can in our model be accounted 10-4 10-3 for by the variation of just a single parameter (b). The most γ· important feature of the frictional phase diagram is reentrant FIG. 6. Flow curves from the stress-controlled simulations. The flow and a critical jamming point at finite stress. The fragile unstable branches (decreasing stress) are obtained as time averages “shear jammed” states observed in the experiments [11] then over the transient flow (right axis: stress values used for the time correspond to the reentrant (inertial) flow regime in our series in Fig. 3). theory. Our work allows one to bring together previously conflicting results [6–8,10] and opens a new path towards a metastable states), we have tried to construct the flow curves theoretical understanding of a unified jamming transition that in this regime by the following procedure: The monitored encompasses both frictionless as well as frictional particles. time series are truncated as soon as the system jams. The (transiently) flowing part of the time series is averaged over We thank Till Kranz for fruitful discussions. We gratefully time, giving rise to the flow curves, shown in Fig. 6. These flow acknowledge financial support by the DFG via FOR 1394 and curves show clearly a nonunique relation σ(˙γ ) or equivalently the Emmy Noether program (He 6322/1-1).

[1] P. Chaudhuri, L. Berthier, and S. Sastry, Phys. Rev. Lett. 104, [13] A. Lees and S. Edwards, J. Phys. C: Solid State Phys. 5, 1921 165701 (2010). (1972). [2]M.P.Ciamarra,P.Richard,M.Schroter,¨ and B. P. Tighe, Soft [14] See Supplemental Material at http://link.aps.org/supplemental/ Matter 8, 9731 (2012). 10.1103/PhysRevE.89.050201 for details of the estimation of [3] C. Song, P. Wang, and H. A. Makse, Nature (London) 453, 629 the packing fractions and further information about the model. (2008). [15] P. D. Olmsted, Rheol. Acta 47, 283 (2008). [4] F. da Cruz, S. Emam, M. Prochnow, J.-N. Roux, and F. Chevoir, [16] C. B. Holmes, M. E. Cates, M. Fuchs, and P. Sollich, J. Rheol. Phys. Rev. E 72, 021309 (2005). 49, 237 (2005). [5] M. Trulsson, B. Andreotti, and P. Claudin, Phys.Rev.Lett.109, [17] P. Olsson and S. Teitel, Phys. Rev. Lett. 109, 108001 118305 (2012). (2012). [6] S. Chialvo, J. Sun, and S. Sundaresan, Phys. Rev. E 85, 021305 [18] B. P. Tighe, E. Woldhuis, J. J. C. Remmers, W. van (2012). Saarloos, and M. van Hecke, Phys. Rev. Lett. 105, 088303 [7] E. Aharonov and D. Sparks, Phys. Rev. E 60, 6890 (1999). (2010). [8] M. Otsuki and H. Hayakawa, Phys.Rev.E83, 051301 (2011). [19] J. Dhont and W. Briels, Rheol. Acta 47, 257 (2008). [9] M. Otsuki and H. Hayakawa, Phys.Rev.E80, 011308 (2009). [20] M. E. Cates, D. A. Head, and A. Ajdari, Phys. Rev. E 66, 025202 [10] M. P.Ciamarra, R. Pastore, M. Nicodemi, and A. Coniglio, Phys. (2002). Rev. E 84, 041308 (2011). [21] H. Nakanishi, S.-i. Nagahiro, and N. Mitarai, Phys. Rev. E 85, [11] D. Bi, J. Zhang, B. Chakraborty, and R. Behringer, Nature 011401 (2012). (London) 480, 355 (2011). [22] The actual transition (“binodal”) has to lie somewhere in the [12] The variation of μ has been studied by Otsuki et al. (see shaded region. Its location cannot be constructed within the Figs. 9 and 15 and the related discussion in [8]). Near μ = 2 current model (see [19]). the rheological properties depend weakly on this parameter and [23] C. Heussinger and J.-L. Barrat, Phys. Rev. Lett. 102, 218303 μ = 2 can be regarded as the limit of large friction. (2009).

050201-4 Supplemental Material: Jamming of Frictional Particles: a Non-equilibrium First Order Phase Transition

Matthias Grob,1 Claus Heussinger,2 and Annette Zippelius1,2 1Max Planck Institute for Dynamics and Self-Organization, Am Faßberg 17, 37073 Göttingen, Germany 2Institute for Theoretical Physics, Georg-August University of Göttingen, Friedrich-Hund Platz 1, 37077 Göttingen, Germany (Dated: April 17, 2014)

I. ESTIMATION OF THE PACKING 0.001 FRACTION (ϕc,ϕσ,ϕη)

0.0008 This section describes the estimation of the packing ϕ < ϕ fractions (ϕc, ϕσ, ϕη) by molecular dynamics data. The 0.0006 c ﬂow curves in the main article Fig. 1 show a jump in ´ stress for ϕ = 0.795 and smooth behaviour for φ = γ 0.0004 ϕ 0.7925. Therefore we conclude that φc lies in between c these two values. This is furthermore in agreement with 0.0002 Fig. 4 in the main article where the ﬁrst solid states ϕη ϕσ (τjam

This section puts emphasize on the packing fractions (ϕc, ϕσ, ϕη) in the framework of the simple model and aims to provide a better understanding of the phase diagram (schematic) in Fig. 5 in the main article. Fig. 2 is a log-linear representation of the ﬂow curves shown in the main article Fig. 1. The black line is subcritical

10-2 Fit Fit Data Data 10-4

-1 -3 ´ γ 10 η and crosses smoothly from plastic to inertial flow as the strain rate is decreased. Upon an increase in packing 10-5 10-4 10-4 10-3 10-2 10-2 5´ 10-2 fraction, ϕc is the first flow curve with zero slope (analog ϕσ - ϕ ϕη - ϕ to the critical isotherm in classical theory of first order phase transition for a fluid-gas-system). At ϕσ the ze- roline (˙γ = 0) is touched at finite stress and at ϕη the FIG. 1: Left: Fit of the strain rateγ ˙ σ, where the plastic coefficient a = 0 which makes the negative (linear) term α stress in the upper branch drops to zero toγ ˙ σ (ϕσ ϕ) , dominant for σ 0. Therefore for ϕ and above the ∝ − → η yielding ϕσ = 0.8003(2) and α = 1.0(1). Right: Fit of the flow curves bend downwards at the origin, i.e. the in- 1 4 inverse generalized viscosity to η− (ϕη ϕ) yielding ϕη = ∝ − ertial branch is vanished, which is reflected in a yield 0.819(1). stress. 7. Shear rheology of frictional grains

7.2. Rheological chaos of frictional grains

Reprinted article with permission from Grob, Matthias and Zippelius, Annette and Heussinger, Claus Physical Review E 93 030901 (2016) http://dx.doi.org/10.1103/PhysRevE.93.030901 Copyright (2016) by the American Physical Society.

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Rheological chaos of frictional grains

Matthias Grob, Annette Zippelius, and Claus Heussinger Institute of Theoretical Physics, Georg-August University of Gottingen,¨ 37073 Gottingen,¨ Germany (Received 28 July 2015; revised manuscript received 24 February 2016; published 14 March 2016) A two-dimensional dense fluid of frictional grains is shown to exhibit time-chaotic, spatially heterogeneous flow in a range of stress values, σ , chosen in the unstable region of s-shaped flow curves. Stress-controlled simulations reveal a phase diagram with reentrant stationary flow for small and large stress σ . In between, no steady flow state can be reached, instead the system either jams or displays time-dependent heterogeneous strain ratesγ ˙(r,t). The results of simulations are in agreement with the stability analysis of a simple hydrodynamic model, coupling stress and microstructure which we tentatively associate with the frictional contact network.

DOI: 10.1103/PhysRevE.93.030901

Discontinuous shear thickening is a ubiquitous phe- remains at rest. In shear direction we use periodic boundary nomenon, observed in many dense suspensions [1,2]. Sim- conditions. ulations for non-Brownian particle suspensions [3–5]as Constitutive equation. Previous work [6,7] has revealed well as for frictional granular media [6,7] have highlighted discontinuous shear thickening for a range of packing fractions the particular role played by frictional particle interactions. close to the jamming transition. The flow curves for frictional Discontinuous shear thickening implies a region of shear stress granular particles are well represented by the following which (at finite Reynolds number, Ref. [8]) is not accessible constitutive equation: to a homogeneous system in stationary state. What happens if the system is forced into this regime by prescribing the γ˙(σ) = aσ1/2 − bσ + cσ 2, (1) stress at the boundary in the unstable region? One possibility is vorticity banding [9], corresponding to bands with different where the term −bσ is due to the frictional interactions, and stress values at the same shear rate. However, there is no clear dependence on the packing fraction φ is implemented with evidence for persistent vorticity banding in experiment so far. a = a(φ)[7,12]. This gives rise to the phenomenology of Furthermore objections have been raised as to the possibility the van der Waals theory for a first-order phase transition: ∼ of vorticity banding as a stationary state: The pressure—in jamming first occurs at the critical point φc = 0.795 which contrast to the shear stress—has to be the same across the also marks the onset of hysteresis. A finite yield stress ∼ interface of the bands; otherwise particle migration is expected first appears at φσ = 0.8003, while the generalized viscosity, 2 ∼ to occur and thereby destabilize the interface. If stationary η = σ/γ˙ , diverges only at φη = 0.819. A similar, but not states are not accessible to the system, we expect to observe identical, sequence of characteristic packing fractions has time-dependent, inhomogeneous states, either oscillatory or been proposed in [6,13]. Most remarkable is the existence chaotic [10]. (due to the b-dependent term) of a regime of shear stress ∂γ˙ In this Rapid Communication we show that spatiotemporal which is unstable ∂σ < 0, corresponding to s-shaped flow chaos occurs in a two-dimensional system of frictional curves. In strain-controlled conditions such an s shape leads granular particles subject to an applied stress which is chosen to discontinuous shear thickening. In the van der Waals theory in the unstable region of the flow curve. We present results it corresponds to the coexistence region. from simulations and formulate a hydrodynamic model to Simulations in the unstable regime. These s-shaped flow derive a phase diagram and identify the regions of parameter curves are indeed observed in the simulations (see Fig. 1), space, where time-chaotic, inhomogeneous solutions are to be however, only as transients and only in rather small simulation found. cells (N<24 000). In larger systems the s shape is slowly We simulate a two-dimensional system of N soft, frictional eroded and vanishes completely above a certain system size. particles in a square box of linear dimension L as detailed Instead a regime of continuous shear thickening develops. in [7]. The particles all have the same mass m = 1, but Closer inspection of the simulations in this regime reveals are polydisperse in size with diameters 0.7, 0.8, 0.9, and that no simple steady state is reached. Rather, the system 1 in equal amounts. Normal and tangential forces, f (n) and displays time-chaotic and spatially inhomogeneous behavior. f (t), are modeled with linear spring-dashpots of unit strength An example is shown in Fig. 2, which is a sequence of for both elastic and viscous contributions. Thereby units of four snapshots of the local-stress field (movies are given in time, length, and mass have been fixed [11]. Flow curves for the Supplemental Material [14]). The system does not settle other viscoelastic parameters are presented in the Appendix. into a time-independent steady state on the time scale of the Coulomb friction is implemented with friction parameter μ = simulations. Instead one observes time-dependent large-scale 2, corresponding to the high friction limit. We expect the same structures, e.g., shear bands which seem to propagate in qualitative findings as presented in this Rapid Communication the principal stress direction, alternating with approximately for all values of μ>0 and refer to a systematic study of the homogeneous states and random large-scale structures. In μ dependence in Ref. [6]. In the stress-controlled simulations, Fig. 5 we show the corresponding time-dependent strain a boundary layer of particles is frozen and the boundary at rate. One clearly observes irregular time dependence with the top is moved with a force σLeˆx , whereas the bottom plate intermittent oscillatory periods.

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∗ -2 relaxes to a stationary state ∂t w = (w − w )/τ which should 10 ∗ vanish in the absence of stress w (σ → 0) → 0. Furthermore dynamic rearrangements occur only due to driving, so that τ −1 ∝ γ˙. So far the model is the same as considered by 10-3 Nakanishi et al. [16] for dilatant ﬂuids. However, the coupling of stress and microstructure is different for the frictional

σ grains under consideration. The frictional contacts reduce LE, N=80000 the ﬂow and hence the velocity gradient. Starting from the 10-4 σ fixed, N=32000 constitutive equation for the strain rate of frictionless grains 1/2 2 =24000 γ˙0 = aσ + cσ , we take the strain rate of our system of =16000 particles with friction to beγ ˙ = γ˙0 − w. This completes the =12000 deﬁnition of the hydrodynamic model -5 10 = 2 0 0.0002 0.0004 0.0006 0.0008 ∂t γ˙ ∂y σ,

γ· γ˙ = γ˙0 − w,

γ˙ ∗ FIG. 1. Evolution of flow curves σ (˙γ ) with system size N.For ∂t w =− (w − w ). (2) small systems a pronounced s shape is visible in transients before the = system jams. In large systems no steady flow occurs; average over Here we have introduced a proportionality constant, τ /γ˙, the time-dependent flow results in continuous shear thickening (stress which can be fitted, when comparing simulations to the controlled except for N = 80 000). predictions of the model. For the stability analysis, it is irrelevant. Higher-order diffusive terms may be added to the stress and/or the w equation. We have checked that the Hydrodynamic model. To understand these time-dependent inclusion of such terms does not change the stability analysis. solutions and locate the regions of parameter space where Other hydrodynamic models of granular fluids include velocity they can occur, we now formulate a hydrodynamic model, fluctuations [17], e.g., granular temperature which, however, determine its stationary states, and analyze their stability. cannot explain effects due to friction. Our starting point is the momentum conservation equation The model allows for two stationary states: The first one in the form ∂t vx = ∂y σxy. For simplicity we only consider corresponds to stationary flow and is explicitly given by a one-dimensional model, allowing for a velocity v in the x = − ∗ = ∗ = flow direction, dependent on y only. In addition we introduce γ˙ γ˙0 w ,w w ,σ σ0. (3) a variable w(y,t) for the internal state or the microstructure Given that w∗ should vanish for vanishing shear, we take of the fluid. Such a variable has been introduced for many it as w∗ = bσ, so that we recover the constitutive relation complex fluids, such as liquid crystals, entangled polymer for frictional grains, Eq. (1). The second stationary solution solutions, and colloidal systems [9]. Here we associate it accounts for the jammed state and reads with the frictional contact network. In the simplest model = = = we only consider a scalar variable, representing, e.g., the γ˙ 0,w γ˙0,σ σ0. (4) number of frictional contacts [8,15], but are aware that For both stationary states, the stress σ = σ0 is homogeneous a tensorial quantity, such as the fabric tensor, might be over the sample, in agreement with the Navier-Stokes equation more appropriate. We assume that the microstructure variable which require a homogeneous stress in two dimensions and hence does not allow vorticity banding. Stability analysis. To study the stability of the stationary states, we consider small deviations δσ,δw ∼ e t eiky and linearize Eq. (2)inδσ,δw. As expected the stationary flow ∂γ˙ is unstable for ∂σ < 0. Below φc, this does not occur and 2 we find two stable modes: a hydrodynamic one, 1 ∝−k , corresponding to the conservation of momentum, and a ∝−∂γ˙ → nonhydrodynamic one, 2 ∂σ 0, corresponding to the relaxation of the microstructure, whose relaxation time becomes infinite as φ → φc and σ → σc. Above φc, the model predicts two stable stationary flow solutions: inertial flow at small stress and plastic flow at large stress. In between, a gap of unstable stress values occurs such that no stationary homogeneous flow is possible in this range of stresses. A typical flow curve in the range φc <φ<φσ is shown in the inset of Fig. 3 as line 1, indicating the unstable regime as red. The jammed state is only stable for φ>φσ in the region where the constitutive relation yields a negativeγ ˙. A typical FIG. 2. Four snapshots of the local stress, revealing large-scale, flow curve in the range φσ <φ<φη is line 2 in the inset time-dependent structures; φ = 0.8035. of Fig. 3 with the stable jammed state marked in blue. For

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plastic flow · 2 γ

1 -4 10 (a) jamming 1 σ σ 2 -3 10 (b) time dependent 10-4 0 γ· inertial flow 10-5 0 φ φ φ c σ η φ 0 20000 40000 60000 80000 100000 t FIG. 3. Inset: Typical stationary states (flow curves) in the range φc <φ<φσ (1) and φσ <φ<φη (2). Unstable regions are FIG. 5. Strain rateγ ˙(t) as a function of time for several values of highlighted in red (time-dependent flow) and blue (jamming). Main: stress, corresponding to path 1 (top, φ = 0.7975) and path 2 (bottom, Phase diagram following from the linear stability analysis; the two φ = 0.8035) in Fig. 3; the sudden drops of the strain rate in the time- generic flow curves, displayed in the inset, correspond to the paths, dependent flow curve indicate that the system nearly gets jammed. denoted by 1 and 2.

shown in Fig. 5(a) for three different stress values. The lowest φ>φη, only stationary plastic flow and stationary jamming are predicted by linear stability analysis. The resulting phase one shows stationary flow in the Bagnold regime, the chaotic diagram of the model is shown in the main panel of Fig. 3. time dependence with oscillatory components is represented Above the critical packing fraction, φ , a finite range by the two red curves for intermediate stress, and larger stress c gives rise to stationary plastic flow [not shown in Fig. 5(a)]. of inaccessible stress values with ∂γ˙ 0 exists and gives ∂σ Similarly in Fig. 5(b) we show the strain rate as a function of rise to a corresponding range of unstable wave numbers time for path 2. In addition to the steady-state flow, and the k2 k2 =|γ˙ ∂γ˙ |, which shrinks to 0 as φ → φ . Hence the c ∂σ c oscillatory flow, there are stationary jammed states, where the model predicts an approximately harmonically oscillating state initial flow ceases after a certain amount of time. at the onset of instability, while well inside the unstable region To get a quantitative measure of the irregularity in the more and more wave numbers are unstable so that one expects time dependence of the states, we have computed the power a broad range of frequencies to be present in the spectrum. spectrum C(ω) as the Fourier transform of the strain rate These expectations are born out by numerical integration of autocorrelation function (Fig. 6). In the stable regimes, the the partial differential equations in order to obtain the full power spectrum is C ∼ ω−2, suggesting simple exponential nonlinear dynamical evolution. Close to φc, the oscillations correlations and linear noise. In the unstable regime this are approximately harmonic, while we find oscillating and background spectrum is superposed by additional and irregular seemingly chaotic solutions at larger packing fractions (see complex structures. This is a strong indication of truly Fig. 4). nonlinear chaotic dynamics. Noteworthy is also the strong Comparison with simulations. To check the predictions of peak at higher stresses. This corresponds to the fast oscillations the above analysis, we performed stress-controlled simulations visible in Fig. 5(a). (for technical details see Ref. [7]) along paths 1 and 2 in the Finite-size effects. The snapshots in Fig. 2 indicate the phase diagram. The time-dependent strain rate along path 1 is buildup of large-scale coherent structures. A large correlation length has also been observed in Ref. [3] in the context of continuous shear thickening in non-Brownian particle suspensions. These correlations are also in line with the strong · γ finite-size effects observed in the flow curves of Fig. 1. Indeed, -4 in lowering the system size, the oscillatory state acquires 10 a finite lifetime and the system jams. This is because in σ=1.0x10-3 small systems there is a finite probability that large strain → -4 rate fluctuations towardsγ ˙ 0leadtoastablejammed =4.5x10 state [see Fig. 5(b) for an example of such an excursion]. 0 5000 10000 15000 20000 25000 30000 35000 Similarly, the appearance of s-shaped flow curves is due to t system-size-dependent strong strain rate fluctuations towards the jammed state withγ ˙ = 0. FIG. 4. Spatially averaged strain rate vs time from numerically Conclusion. We discuss stress-controlled driving of a gran- integrating the hydrodynamical model for different imposed stress. ular system that undergoes discontinuous shear thickening. In the unstable region at intermediate stress values, we observe In particular a regime is identified where the system does oscillations and chaotic solutions ( = 10−3,φ = 0.7975). not settle into a time-independent steady state. Instead, it

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GROB, ZIPPELIUS, AND HEUSSINGER PHYSICAL REVIEW E 93, 030901(R) (2016)

TABLE I. Normal restitution coefﬁcient (n) and binary collision time t for different k and η.

kη (n) t

1 1 0.305 2.375 1 1/2 0.569 2.26 1 1/10 0.895 2.22 1/2 1 0.163 3.63 2 1 0.44 1.4 10 1 0.7 0.7

with the contacts which are blocked by Coulomb friction, and ) (arb. units)

ω explore the possibility of a tensorial observable, such as the

C( fabric tensor.

Acknowledgments. We gratefully acknowledge ﬁnancial support by the Deutsche Forschungsgemeinschaft via programs FOR 1394 and Emmy Noether He 6322/1-1.

APPENDIX

∝ω-2 In this Appendix we deal with flow curves of frictional grains at fixed packing fraction when the particles’ elastic 10-4 10-3 10-2 10-1 and viscous damping constants change. In the main text ω we use unit strength for elastic and viscous contributions in the linear-spring dashpots (k and η, respectively). Units FIG. 6. Power spectrum C(ω) of strain rate fluctuations at of time and energy dissipation when particles interact are constant stress from low (bottom) to high (top) values of σ (shifted set by this choice. In particular, the coefficient of (normal) vertically for clarity of presentation); in the stable region C ∼ ω−2; restitution, (n), and the binary collision time, t, are set. Table I whereas in the unstable region additional complex structures are summarizes the normal coefficient of restitution and the binary superimposed on the ω−2 decay. collision time for the parameters that we discuss here. We tune both contributions, k = k(n) = k(t) and η = η(n) = η(t), independently. Both (n) and t just serve as a guidance on displays spatiotemporal oscillations and chaotic behavior as properties of pairwise collisions and do not respect the large a novel possibility to adopt to stress in the unstable parts of the packing fraction. Also the important frictional contribution flow curve. Recent experiments on corn starch reveal very which implies tangential restitution (t) is not characterized by similar unsteady flow where theory predicts discontinuous these quantities. The normal restitution can be computed easily shear thickening [18]. while the tangential part is of rich and complicated behavior Simulations reveal a phase diagram that has a characteristic due to its dependence on the impact velocities [19]. re-entrant form with steady-state flow at small and large stresses. At intermediate values of stress either time-dependent states are observed or the system settles into a nonflowing 10-1 jammed state, depending on stress and packing fraction. For 10-2 φc <φ<φσ only time-dependent solutions are observed, -3 whereas for φσ <φa sequence of inertial flow, chaotic flow, 10 jammed state, and plastic flow are seen for increasing values 10-4 of σ , until at φ = φη only a transition from the jammed state -5 / k 10 to plastic flow remains. σ -6 We also present a hydrodynamical model, coupling stress 10 k=0.5 to a microstructural observable. Within linear stability analysis 10-7 =1 we recover the detailed features of the phase diagram as =2 10-8 obtained from simulations. In the unstable region, the model =10 predicts either oscillating or time-chaotic flow. 10-9 -6 -5 -4 -3 -2 -1 In future work we plan to quantify the spatiotemporal 10 10 10 10 10 10 correlations that are visible in the snapshots and compare γ· / k1/2 them with length scales determined in [3] from velocity correlations. We furthermore aim to better understand the FIG. 7. Scaled flow curves for φ = 0.80, viscous damping pa- microstructural observable, check whether it can be associated rameter η = 1 and varying elastic constant k.

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RHEOLOGICAL CHAOS OF FRICTIONAL GRAINS PHYSICAL REVIEW E 93, 030901(R) (2016)

10-1 10-1 10-2 10-2 10-3 10-3 10-4 10-4 σ σ 10-5 10-5 ϕ -6 =0.7925 -6 10 =0.7975 10 η=1 10-7 =0.80 10-7 =1/2 =0.8025 =1/10 10-8 10-8 10-6 10-5 10-4 10-3 10-2 10-1 10-6 10-5 10-4 10-3 10-2 10-1 γ· γ·

FIG. 8. Flow curves for varying packing fraction φ across the FIG. 9. Flow curves for φ = 0.80, elastic constant k = 1, and transition with viscous damping parameter η = 1 and elastic constant varying viscous damping parameter η. k = 1.

The following data show flow curves of a system with A systematic study of variations of all three parameters, φ,k, N = 8000 particles. Figure 7 shows scaled flow curves for and η, is out the scope of this article. In Fig. 8 we show a fixed φ = 0.80 and η = 1. When k decreases ( (n) decreases), choice of packing fractions with phenomenology similar to the discontinuity shifts towards small strain rate until it is Fig. 7.Lowerφ leads to flow curves similar to those with missed out by our simulation. Note that the numerical effort large (n), and larger φ shows the same behavior as small (n). for low strain rate is much larger than for large strain rate. An The variation of the viscous damping parameter η is shown in increasing k (increasing (n)) shifts the discontinuity towards Fig. 9. The phenomenology is the same as above: decreasing larger strain rate which makes the discontinuity smaller until it (n) (larger η) leads to a shift towards small strain rates and vanishes. At k = 10 the transition from inertial to plastic flow an increasing coefficient of restitution (smaller η) shifts the is smooth and without shear thickening. The phenomenology discontinuity towards larger strain rate until it vanishes. Then of varying k seems in that range similar to a change of the the transition between inertial and plastic flow is smooth and packing fraction. The latter was studied in previous work [7]. without shear thickening again.

[1] E. Brown and H. M. Jager,¨ Rep. Prog. Phys. 77, 046602 (2014). cannot be evaluated analytically. For a discussion, see J. Schafer,¨ [2] Z. Pan, H. de Cagny, B. Weber, and D. Bonn, Phys. Rev. E 92, S. Dippel, and D. E. Wolf, J. Phys. I 6, 5 (1996).

032202 (2015). [12] These values correspond to a = a0|φη − φ|(φη − φ), a0 = = = 3 2/3 1/3 [3] C. Heussinger, Phys Rev. E 88, 050201(R) (2013). 52.05, c 147.27, and b 2 a(φc) c . [4]R.Seto,R.Mari,J.F.Morris,andM.M.Denn,Phys. Rev. Lett. [13] M. P. Ciamarra, R. Pastore, M. Nicodemi, and A. Coniglio, 111, 218301 (2013). Phys. Rev. E 84, 041308 (2011). [5] N. Fernandez, R. Mani, D. Rinaldi, D. Kadau, M. Mosquet, [14] See Supplemental Material at http://link.aps.org/supplemental/ H. Lombois-Burger, J. Cayer-Barrioz, H. J. Herrmann, N. D. 10.1103/PhysRevE.93.030901 for movies of time-dependent Spencer, and L. Isa, Phys. Rev. Lett. 111, 108301 (2013). flow. [6] M. Otsuki and H. Hayakawa, Phys. Rev. E 83, 051301 (2011). [15] M. Wyart and M. E. Cates, Phys. Rev. Lett. 112, 098302 [7] M. Grob, C. Heussinger, and A. Zippelius, Phys.Rev.E89, (2014). 050201(R) (2014). [16] H. Nakanishi, S. I. Nagahiro, and N. Mitarai, Phys.Rev.E85, [8]R.Mari,R.Seto,J.F.Morris,andM.M.Denn,Phys.Rev.E 011401 (2012). 91, 052302 (2015). [17] N. V. Brilliantov and T. Poschel,¨ Kinetic Theory of Granular [9] P. Olmsted, Rheol. Acta 47, 283 (2008). Gases (Oxford University, Oxford, 2004). [10] M. E. Cates, D. A. Head, and A. Ajdari, Phys. Rev. E 66, [18] M. Hermes, B. M. Guy, G. Poy, M. E. Cates, M. Wyart, and 025202(R) (2002). W. C. Poon, arXiv:1511.08011. [11] This leads to a normal coefficient of restitution of (n) =∼ 0.3. [19] V.Becker, T. Schwager, and T. Poschel,¨ Phys. Rev. E 77, 011304 Due to the frictional interaction the full coefficient of restitution (2008).

030901-5 7. Shear rheology of frictional grains

7.3. Unsteady rheology and heterogeneous ﬂow of dry frictional grains

52 Unsteady rheology and heterogeneous ﬂow of dry frictional grains

Matthias Grob, Claus Heussinger, and Annette Zippelius Institute of Theoretical Physics, Georg-August University of Göttingen,37073 Göttingen,Germany (Dated: June 27, 2016) Using molecular dynamics simulations we investigate heterogeneous and unsteady states in shear thickening frictional granular media. We characterize the heterogeneous states by the decay of the stress-stress correlation function providing a characteristic length scale. In only large simulation cells, in which shear thickening is continuous, both, heterogeneities and unsteady states, are shown to exist. The analysis of stress and strain controlled simulations in differently sized simulation cells explains the role played by a length that is considerably larger than the particle size. Simulations of small simulations cells relate shear jamming to discontinuous shear thickening. Local measures, e.g., the nonaffine velocity or the local coordination number, are shown to follow instantaneously the evolution of stress.

I. INTRODUCTION damentally.

Flow properties of dense granular matter are of funda- A remarkable phenomenon in frictional systems is mental importance in a wide range of engineering disci- shear thickening, i.e., the growth of viscosity with shear plines and nature. Many industrial resources are shipped rate or shear stress. Shear thickening can be continuous and processed as grains. In nature, sand is one of the or discontinuous. Increasing the packing fraction can lead most abundant materials on earth. Collections of grains from a continuous shear thickening scenario to a discon- can also attain solid states when they jam into unordered tinuous shear thickening transition and finally to (shear) and rigid states at large packing fractions. The transition jamming [17, 18]. Flow curves of shear thickening solu- from fluid to solid is called jamming. Granular matter is tions include regions that are mechanically unstable and considered athermal as the energy associated with motion not accessible by homogeneous flow [19]. Moreover, het- or contact is by orders of magnitude above the thermal erogeneous steady states are not possible due to particle energy [1, 2]. The phase diagram for athermal granular migration [16]. Thus, the only remaining flow scenario is media shows that the packing density and the shear stress heterogeneous and unsteady flow. control whether arrest or flow is achieved [3]. Arrest and flow are separated by the yield stress line, which goes to zero stress, Σ = 0, at a packing fraction, φJ, called In this article, we perform numerical simulations to jamming point. For frictionless spheres — spheres which study heterogeneous flowing states in dry granular sys- interact only along the normal direction at the contact tems. These states lead to unsteady flow and continu- — flow and arrest around the jamming point have been ous shear thickening. The heterogeneous states possess investigated thoroughly [4]. a typical length scale which rules out heterogeneous flow However, interactions of grains are typically frictional. in small systems. Instead, small systems shear thicken Rheometry with suspensions of PMMA and silica spheres discontinuously. The discontinuous shear thickening in highlight the role of frictional contacts [5–7]. Experi- small systems appears across an interval of stress. This ments with photoelastic discs show that jamming is pos- interval of stress is identified as unstable. Unstable values sible below the isotropic jamming point when a finite of stress lead to unsteady flow in large systems contrast- shear stress is applied [8]. Corn starch experiments show ing shear jamming in small systems. The overall dynam- a shear induced rigidity transition that is either shear ics is not reflected only in the shear stress but also in thickening or shear jamming [9, 10]. In computer simula- other (local) measures which adopt to the flowing state, tions friction is controlled parametrically, thereby allow- either stationary or unsteady. ing for a direct comparison to the frictionless scenario. Numerical investigations show that φJ loses its exceptional role in the characterization of the rheology close to The article is organized as follows: First, we recapitu- the jamming transition. Instead, more than one packing late in sec. II the description of frictional particle flow by fraction is necessary for a thorough description of flow means of flow curves. In sec. III, we present the molecular and jamming [11–14]. Moreover, simulations with fric- model, our simulation set-up and measured quantities. In tional spheres exhibit novel phenomena in the dense flow sec. IV, we characterize heterogeneous states, show their regime, e.g., transient flow that ceases after a character- time dependence. We relate discontinuous shear thicken- istic time [11, 13], hysteresis in the flow curve [12, 13], ing to shear jamming and continuous shear thickening to the coexistence between a low viscosity and a high vis- time dependent states. Subsequently, we show that the cosity state [15], or even rheo-chaotic response [14]. The width of the velocity distribution shows non-monotonic latter is also found in corn starch experiments [16]. Thus, behavior and is related to the generalized viscosity. A friction influences the dense flow close to jamming fun- discussion and conclusion close the article in sec. V. 2

flow curves look well defined but, in fact, the unstable -2 stress interval reveals time dependent heterogeneous flow 10 which is a subject of this article.

10-3 III. MODEL AND METHODS -4

Σ 10 A. Molecular model 10-5 Σ fixed, N=8000 Σ fixed, N=80000 The setup in this article is identical to those of for- 10-6 LE, N=8000 mer studies [13, 14]. Due to the brevity of the former LE, N=80000 articles we give full details of modeling and the simula- 10-5 10-4 10-3 tion here. We consider N soft particles in a square box γ· of linear dimension, L. N and L determine the packing N 2 2 fraction, φ = i=1 π/4di /L . Forces between particles, FIG. 1. Flow curve for fixed φ = 0.7975 but varying system i and j, arise at contact and are frictional. Frictional con- P (n) size. Imposed strain rate indicated by “LE”. tact forces have a normal component, fij , and a tangen- (t) tial component, fij . We implement the model by Cun- dall and Struck [22] with an additional tangential viscous II. FLOW CURVES OF DRY GRANULAR component as done by others, see, e.g., [12, 21, 23–25]. PARTICLES The Coulomb criterion limits the amount of tangential force by a material specific multiple, µ, of the normal Dry granular media show a rich phenomenology when force: deformed and flow curves (see, e.g., [13, 20, 21]) char- (t) (n) acterize the flow regimes. Below jamming at low strain f µ f . (1) | ij | ≤ | ij | rates, inertial flow (also called Bagnold flow) is characterized by a quadratic dependence of the global stress µ is called the friction coefficient. A contact is said to 2 (t) (n) on the strain rate: Σ γ˙ . In this regime the only rele- slide if fij = µ fij . ∝ 1 | | | | vant time scale is the inverse strain rate,γ ˙ − , dominating The normal interaction is a standard visco-elastic its counterpart, √k 1, set by the spring constant of the model where the overlap, δ(n) = R +R r r > 0, of − ij i j −| i − j| particles, k, which reflects the particle’s stiffness. Bag- particles with radii Ri,Rj at position ri, rj, determines nold flow corresponds to flow of hard spheres. However, the interparticle normal force granular particles possess a finite stiffness that becomes relevant at a characteristic strain rate or when the system (n) (n) (n) (n) ˙(n) fij = k δij η /meffδij ˆnij. (2) is dense: plastic flow, either quasistatic (Σ = const.) or − − 1/2 rapid (Σ γ˙ ), establishes. In previous work [13, 14], The unit vector, ˆn , points from particle i’s to particle the authors∝ suggested s-shaped flow curves which capture ij j’s center. When η(n) > 0 collisions are inelastic and these flow regimes. Implications of the s-shape are flow if contacts are frictionless a coefficient of (normal) resti- instabilities, which eventually lead to chaotic behavior tution can be evaluated analytically: 0.3 with unit in large systems [14], or shear jamming in small systems spring and viscous constants. ≈ [13]. However, the relation between these phenomena The non-sliding tangential force is analog to the nor- remained elusive. mal force. Two particles rotate with angular velocities, Figure 1 shows flow curves for differently sized systems ωi and ωj, and move with velocities, vi and vj, respec- and either controlled by strain rate or by shear stress. In tively. The relative tangential velocity of spheres in con- the shear rate controlled setting, flow curves show shear tact is given by the sum of contributions by rotation and thickening. This is either discontinuous, when the stress tan ˆ translation, vij = Riωi + Rjωj + (vi vj)tij, with the jumps, or continuous when the stress changes continu- ˆ − ously with the strain rate. When the shear stress is con- tangential vector, tij. The tangential viscous force is (t) tan trolled and the strain rate is measured, the discontinuous given by η /meffvij and a restoring contribution is − tan shear thickening turns into an s-shape, which is caused determined by the integral of vij but respecting eq. 1. by excursions of the system to nearly jammed states [14]. (t) (t) tan (t) (t) (t) (n) Explicitly, fij = η vij k δij if fij µ fij . The unstable stress interval in which no persistent flow t+∆t − − | | ≤ | | Where vtan(t )dt is the increment of the tangen- is possible is clearly identified in the small system: the t ij 0 0 (t) stress jumps across an interval of stress when the strain tial displacement,R δij , during the interval [t, t+∆t] when rate is controlled and in this interval we report jammed the contact does not slide. If the contact slides, we set (t) (t) (n) (t) (t) (t) states when the stress is controlled. The large system’s δij = µ/k fij and consequently fij = k δij to sat- characterization is subtle: irrespective of the protocol, isfy Coulomb’s| criterion.| 3

ai and bj, of particles, i and j, as: 1 C (r = (x, y)) = a b δ(x x)δ(y y). (4) ab N i j ij − ij − 1.00π i=j X6 As correlation functions along directions v we com- 0.75π 1,2 pute: ϑ1 ϑ1 ϑ 0.50π 1,2 1 1 2 ϑ2 C (r) = a b δ(x r cos ϑ )δ(y r sin ϑ ). ab N i j ij − 1,2 ij − 1,2 ϑ2 i=j 0.25π X6 (5) y 0.00π Averages are distinguished between spatial and tem- 10-4 10-3 10-2 poral averages. The spatial average of a quantity, x, over x γ· a sample with N particles reads: 1 N x = x . (6) FIG. 2. Left: Simple shear geometry implies a compression h iN N i i=1 and dilation direction with defining angles, ϑ1/2. Right: Com- X pression and dilation direction as a function of strain rate The corresponding standard deviation is defined as (φ = 0.7975,N = 80000). 2 2 ∆N x = x N x N . The temporal mean is defined with N measurementsh i − h i at t with s 1, ..., N as Tp s ∈ { T } N Following previous studies [12–14], four differently 1 T sized particle species populate the system with equal x T = x(ts). (7) h i N amounts and diameters, d = 0.7, 0.8, 0.9, 1.0, which sets T s=1 X the unit of length. The unit of mass is set by the choice The corresponding standard deviation is defined as m = 1. We set µ = 2 and we are aware of the compar- ∆ x = x2 x 2 . atively high value of µ, which corresponds to the high T h iT − h iT friction limit [12]. p IV. RESULTS

B. Protocol, geometry and measured quantities A. Heterogeneous time dependent states

To control the strain rate we use Lees-Edwards bound- In this section, we study heterogeneous flow states that ary conditions (LE). In stress-controlled simulations we we observe in large simulation cells in the regime of con- freeze a bottom layer of particles which remains at rest tinuous shear thickening. We explain that the heteroge- and a boundary layer at the top is moved with a force neous flow states are time dependent as we expect due σLex; periodic boundary conditions are used in shear to the presence of strong gradients in the stress fields. (x-)direction. The geometry is sketched in fig. 2 (left). We focus on the evolution of the stress field and on the Simple shear geometry implies two distinguished direc- global stress time series. tions: compression and dilation direction. These direc- Here, we investigate properties of states in which we tions are parallel to the eigenvectors of the stress tensor: find characteristics of both, inertial and plastic flow, at the same time. Patches of a typical size show signatures of either inertial or plastic flow. We focus on measure- Σ Σ Σ = xx xy , (3) ments of local properties, e.g., the stress of a particle i, Σyx Σyy x y σi = j=i rijfij. Other indicators are the local coordination number6 and the local nonaffine velocity as shown α β P with Σαβ = 1/N i=j rijfij and rij = ri rj. In this in appendix A. Large systems (N > 24000) with inter- 6 − paper we use the short hand notation: Σ = Σxy. The mediate average stress show well separated patches dif- P eigenvectors, v1,2, of Σ enclose angles, ϑ1,2, with the x- fering by orders of magnitude in the absolute value of axis. In fig. 2 (right) we show that these directions are local shear stress, see fig. 3. There are well visible stress orthogonal and independent on the strain rate and thus bands in compression direction that do not percolate in on the flow regime. dilation direction. Remarkably, not everywhere do these In the course of this manuscript we consider correla- stress bands span the system in compression direction. tion functions. To take anisotropy into account we are In dilation direction the shear stress changes dramati- interested in correlations not only dependent on the dis- cally while we see only weak variations in compression tance of particles, r = r , but on the displacement vector, direction. Thus, we can distinguish clearly between com- r = (x, y). We compute| | the correlation of two quantities, pression and dilation direction by visual inspection. 4

3.5 inertial 3 time dependent 2.5 T

> 2 Σ

/ < 1.5 Σ 1

0.5 0 15 16 17 18 19 20 t/γ·-1

FIG. 5. Shear stress as a function of time leading to time 4 dependent ﬂow with large amplitude (green,γ ˙ = 1.25 10− ). For comparison, we show a time series in the inertial× ﬂow 5 regime (purple,γ ˙ = 1 10− ). N = 80000, ϕ = 0.7975. ×

102 FIG. 3. Snapshot of local stress shows heterogeneous states. N=8000 N=80000 φ = 0.7975 and N = 80000 (L 240) with strain rate ofγ ˙ = 4 ≈ 1.25 10− . Particles with zero stress are colored according to the× lowest nonzero value that is depicted. T > Σ 101 / < Σ T ∆

1/2 N

100

10-5 10-4 10-3 10-2 · FIG. 4. Time evolution of local stress with same parameters γ 1 as in fig. 3. Time difference ∆t 0.2γ ˙ − . ≈ FIG. 6. Relative standard deviation of the measured global stress normalized by 1/√N. ϕ = 0.7975. In the snapshot, fig. 3, the local stress differs by orders of magnitude. These strong stress gradients lead to flow/deformation, i.e., the snapshot shown above is the time series in [6, 26]. To quantify the degree of the a realization of a time dependent state. The sequence time dependence, we show in fig. 6 the standard deviation of snapshots in fig. 4 shows the evolution in time (for of the measured global stress, ∆T Σ, for differently sized movies, see the supplemental material to [14]). The stress systems. Asymptotically, for low and large strain rate, band that we see in fig. 3 wanders in dilation direction the fluctuations scale with the square root of the number (fig.4 left), eventually interacts with other forming stress of particles, √N. In the large system fluctuations show 4 3 bands (fig.4 middle) and vanishes. Sequences of almost a broad peak (γ ˙ = 10− 10− ) as an evidence for the − homogeneous states (fig.4 right) follow before new het- time dependence in the shear thickening regime. Over erogeneities emerge. Stress bands form again, wander in this broad range of strain rate, the large system shows dilation direction and the dynamics repeats. a larger standard deviation than the small system which The dynamics of the flowing state leads to a time de- is yet in the plastic flow regime. The peak in the data 4 pendent stress on the particles, σ. This is reflected by the of the small systems atγ ˙ 10− stems from short-lived ≈ global average of shear stress, Σ, shown in fig. 5, where excursions from an inertial flow state to a plastic flow we compare a highly time dependent state (of which in- state (not shown). stants are shown in the snapshots) with steady flow in the In the course of this article, we take averages over inertial flow regime. The strongly time dependent data heterogeneous and unsteady states. We justify this by confirms the expectation of unsteady flow and reminds on the stationarity of the shear stress distribution. As time 5

P(Σ; 0, 100) 1 0.14 10-1 P(Σ; 0, 50) Σ 2 P( ; 0, 25) 0.12 P(Σ; 25, 50) 0.10 t) ∆ 0.08 10-2 ; t, t+

Σ 0.06 C(r)/C(0) P( 0.04 0.02 10-3 0.00 -5 -4 -4 -4 -4 0 5.0x10 1.0x10 1.5x10 2.0x10 2.5x10 0 20 40 60 80 100 Σ r/d

FIG. 7. Distribution of global stress in a specific time window FIG. 8. Stress correlation functions for the heterogeneous as indicated. Data corresponding to the time dependent time time dependent state up to L/2 120. Average in compres- 1 ≈ series shown in fig. 5. sion direction, Cδσδσ(r) , is shown in green and standard deviation is markedh in gray.i Average stress correlation in dilation direction, C2 (r) , is shown in blue. Parameters h δσδσ i progresses, we measure the global stress, Σ(t), and com- according to fig. 3. pute the distribution, P (Σ; t, t+∆t), between times t and t+∆t. When this distribution is independent of t, the distribution is stationary. In fig. 7, we show P (Σ; t, t + ∆t). tances the stress correlation follows the pair correlation The distributions coincide. This confirms the station- function, g(r) (not shown), which gives rise to a series of arity of the stress distribution providing the reason for maxima and minima for r < 4. On long distances the cor- taking averages over finite time intervals. relations decay slowly and exhibit the anisotropy clearly. The average stress correlation in compression direction, The conclusion to this part is that there is only a het- 1 erogeneous unsteady state that leads to an intermediate Cδσδσ(r), does not decay to zero on a distance L/2. This value of stress that gives rise to continuous shear thick- corresponds with the persistence of stress in compression direction that we see in the snapshot (fig. 3). In dilation ening as shown in the flow curve fig. 1. Only averages 2 over unsteady and heterogeneous flow states lead to inter- direction the correlation function, Cδσδσ(r), crosses zero at about r 45 which corresponds roughly to half the mediate stress values which are not accessible by steady ≈ flow. However, we showed that characteristic flow fea- width of the stress band shown in the snapshot. We quantify the decay of the correlation by the length, tures in the dynamics repeat. Moreover, the stress dis- 1 tribution is stationary. The heterogeneity in these states l, where the correlation function, Cδσδσ(r), is decayed to is in stark contrast to the homogeneous states in recent 0.025 [27]. The resulting length is shown as a function of experiments with PMMA particles which also show dis- strain rate in fig. 9. We find that the correlation in com- continuous shear thickening due to frictional contacts [6]. pression direction behaves non-monotonically as a function of strain rate. In the regime of time dependent flow 4 3 (γ ˙ = 10− 10− and N 32000), the correlation length peaks. The− length ratio,≥l/L, plateaus for large system B. Stress correlation and anisotropy at intermediate strain rate, see inset of fig. 9. For lower or larger strain rates, the correlation lengths are compar- As shown in the previous part, heterogeneous flow atively small and similar to each other. The small sys- states are unsteady but show a stationary stress distri- tem does not exhibit heterogeneous and time dependent bution. We make use of this and time average to extract states consistent with the comparatively small correlation characteristic lengths and fluctuations by means of stress length that we measure for N 8000. The correlation correlation functions. We contrast our findings to homo- length in dilation direction (not≤ shown) exhibits the same geneous and stationary flow. qualitative behavior but is smaller than in compression As seen in the sequence of snapshots in fig. 4, the stress direction. field changes its characteristics in course of its evolu- We use an analog of a structure factor of a density tion. However, we are interested in the average char- field to identify contributions to stress correlations with acterization of the heterogeneous time dependent flow. different wave numbers. For a motivation see appendix To quantify the characteristic length and anisotropy we C. Bands with low stress and large stress alternate compute stress correlations as in eqn. 5, with ai = bi = in characteristic large scale structures, see fig. 3. The σi σ N = δσi. The spatial stress correlation averaged large scale structures are associated with contributions over− hseverali snapshots is shown in fig. 8. On short dis- to small wave numbers in the analog of the structure fac- 6

80 40 N=4000 1.0 N=4000 70 =8000 35 =8000 /L 0.5 60 =32000 30 =32000 max =64000 l =64000 50 0.0 25 =80000 0 60 120 180 =80000

=128000 3/4 =128000 l 40 L 20 30 s N 15 20 10 10 5 0 0 10-5 10-4 10-3 10-2 10-5 10-4 10-3 10-2 · · γ γ

FIG. 9. Main: Distance, l, where the correlation function, FIG. 11. Average standard deviation of the stress correlation 1 Cδσδσ(r), is decayed, as a function of strain rate. Inset: Rela- in compression direction, s, as a function of strain rate. The max 4 tive length, l /L, forγ ˙ = 1.58 10− as a function of linear N-dependence is explained in appendix B. system size, L. ×

Now, we want to examine the fluctuations of the stress 0.4 -4 correlation functions. As shown in fig. 8 (gray area), we compute the binwise standard deviations of the stress ) 0.3 compression dilation -5 y correlation to characterize fluctuations in a distance r/L. , k x -1

(k To compare diﬀerent systems by means of a single num-

0.2 -6 σ / d S y ber, we average the standard deviations in compression k 10 0.1 -7 direction over all Nb bins: log

0 -8 Nb 1 1 -0.4 -0.2 0 0.2 0.4 s = std Cδσδσ(ri). (9) -1 N k / d b i x X FIG. 10. The Structure factor for heterogeneous time depen- Figure 11 shows s as a function of strain rate. The av- dent flow exhibits strong anisotropy and shows a large contri- erage standard deviation confirms that the small system bution in dilation direction for small wave numbers. Arrows does not show time dependent behavior while the large indicate compression and dilation direction. Parameters ac- system shows highly unsteady dynamics in the continu- cording to fig. 3. ous shear thickening regime. In this part, we examined average properties of correlations and the structure of the stress field. Even though tor for the stress field: the flow is unsteady, we can distinguish clearly between compression and dilation direction, and, more impor- NS (k) = σˆ(k)ˆσ( k) σˆ(k) σˆ( k) . (8) σ h − iT − h iT h − iT tantly, we report a characteristic length, which is given Figure 10 shows the structure factor of the stress field, by the average decay of the stress correlation function. eq. 8, for time dependent flow. A large contribution Moreover, the fluctuations of the correlation functions (bright) at low wave numbers in dilation direction sig- reflect the dynamics of the heterogeneous unsteady flow. nals the heterogeneity that corresponds to the large scale structure in the snapshot. Only small contributions are found in directions other than dilation direction or for C. Apparent shear thickening and shear jamming larger wave numbers. Only the heterogeneous time dependent regime shows a remarkable peak for the smallest In this section we link results with controlled strain wave numbers which can be resolved. This wave number rate and results with controlled stress together. In small corresponds to the size of the stress band that we see systems, this relates discontinuous shear thickening to in the snapshot (fig. 3) and suggests that the system re- shear jamming. In large systems we find continuous shear quires a certain size to build up heterogeneous and time thickening and time dependent states instead. dependent states. The inertial flow regime for low strain First, we focus on strain rate controlled simulations. rate and the plastic flow regime for large strain rate do The stress that is shown in the flow curve, fig. 1, is a not show long ranged structures (not shown), i.e., the result of time-averaging over possibly metastable or time system is homogeneous. dependent states. Examples of such states are shown 7

0 2 4 6 8 0 0

10-3 10-3

-4 Σ

Σ 10 -4 N=4000 10 =8000 =12000 =16000 10-5 N=80000 =24000 =80000 N=8000 10-5 0 1000 2000 3000 4000 5000 6000 0 10 20 30 40 10-4 10-2 ·-1 τ -1/2 t / γ P(Σ) jam / k

FIG. 12. Left: Smoothed time series, Σ(t), for ϕ = 0.7975. FIG. 13. Time of transient flow, τjam, for fixed φ = 0.7975 but 4 Small system (N = 8000):γ ˙ = 1.125 10− leads to temporal different system size, N = 4000 80000. When the system coexistence of metastable states at low× and large stress. Large never jams, we show the simulation− time (5000). 4 system (N = 80000):γ ˙ = 1.58 10− leads to intermediate but unsteady stress (time at top× axis). Right: Corresponding stress distributions. find shear jamming at this packing fraction but unsteady flow instead. To conclude this part, the results of simulations with in the time series in fig. 12. In green we show a time different protocol and differently sized simulation cells di- series of the small system with an intermediate strain rectly relate to the flow curves shown in fig. 1. Small sys- rate where flow is metastable: Both, large and low stress tems follow a sigmoidal flow curve and show metastable states, are attained alternately. The shear stress jumps and unstable states, i.e., alternating stress at fixed strain between the two states which differ in stress by more rate and shear jamming at fixed shear stress, respec- than an order of magnitude. This jump determines the tively. When the system is large we observe apparent discontinuous shear thickening in the flow curve. The or- shear thickening and we cannot find jammed states at ange curve shows a time series of the large system with this packing fraction in the stress controlled setting. In- an intermediate strain rate leading to apparent continu- stead, unstable flow is realized by time dependent states ous shear thickening in the flow curve. At this strain rate as predicted by the model in [14]. we see a time dependent state. Remarkably, the time dependent state lives in an interval of stress which is not favored by the small system. Instead, the time series and flow curve of the small system exhibit jumps across D. Velocity distributions that interval of stress. The distribution of the metastable state is bimodal with peaks at metastable values of stress. In this section, we investigate the distribution of non- The peak of the distribution for the time dependent state affine velocities. We describe the time evolution and av- resides at the local minimum of the distribution of the erage characteristics in terms of the width of distribution metastable flow. This reminds on the results in [15, 28]. and relate the findings to the flow curves via the gener- Thus there is an unstable band of stress values leading alized viscosity. to jumps in the small system and time dependent flow in In the heterogeneous state, stress varies by orders of the large system. magnitude across the system and these strong gradients Next, we focus on stress controlled simulations. To suggest a broad distribution of velocity, see fig. 19 for probe jamming in the unstable region we impose stress an example of a heterogeneous velocity field. We com- following previous studies [13]. In the unstable regime we pute the velocity distributions in gradient direction over observe that the flow ceases and the system jams when the particles, P (vy), in a single configuration as shown the simulation cell is small. Figure 13 shows the time in fig. 14. By inspection of the distributions we find a of transient flow in differently sized systems. In small remarkable non-monotonicity as a function of the strain systems, N = 4000 16000, we find that the system rate. At large strain rate the width of the velocity distri- jams after a finite transient− time which is characteristic bution is small but increases as the strain rate decreases. for the stress that is applied. Jamming occurs around In the shear thickening regime at intermediate strain rate the minimum in the bimodal distribution in fig. 12. This the distribution is the widest. At low strain rate the dis- is the same range of stress where we find the jumps in tribution is again narrow. For all strain rates the distri- the time series fig. 12 or in the flow curve in fig. 1. In bution is peaked at zero and symmetric due to the prob- contrast, in large simulation cells, N > 24000, we cannot lem’s symmetry. Hence, the width of the distribution is 8

14 100 ϕ = 0.78, N=80000 12 ϕ = 0.7975, N=8000 -1 ϕ = 0.7975, N=80000 10 = 0.8035, N=80000 10 ϕ -2 • ) · γ / • γ 10 γ=.05 / T 8 y =.01

12 v> · -3 =.005 γ N P(v 10 9 v/ =.0016 ∆ 6 ∆ =.0005 6 < -4 3 10 =.00016 4 =.00013 10-410-310-2 -5 =.00005 2 10 =.000025 γ· 10-2 10-1 100 101 0 • 10-4 10-3 10-2 vy/γ γ·

FIG. 14. Velocity distribution of individual snapshots at dif- FIG. 16. Time averaged width of the velocity distributions ferent strain rates for φ = 0.7975 and N = 80000. Inset: showing non-monotonic behavior. Width of the velocity distribution as a function of strain rate.

9 -3 • 10 105 ∆Nv/γ ϕ = 0.78, N=80000 8 Σ ϕ = 0.7975, N=8000 ϕ = 0.7975, N=80000 7 104 ϕ = 0.8035, N=80000 2 • γ 6 · -4 γ v/

10 Σ N 3 / ∆ 5 10 Σ

4 = η 2 3 10

2 10-5 0 1 2 3 4 5 101 •-1 -4 -3 -2 t/γ 10 10 10 γ·

FIG. 15. Time series for the width of the velocity distribu- FIG. 17. Generalized viscosity showing non-monotonic be- tion (left axis, red) and the shear stress (right axis, blue). havior. Parameters according to ﬁg. 3.

a suitable measure for the typical velocity of a particle small systems and continuity as a result of taking the av- in the system. These widths are shown in the inset of erage over time dependent quantities in the large system fig. 14, confirming the non-monotonic behavior. for φ = 0.7975. The standard deviation has a maximum However, the velocity distributions are affected by the near the jump in the small system and in the time depen- unsteady character of heterogeneous flow. At each in- dent regime in the large system. A packing fraction of stant of a simulation, t, we compute P (vy, t) individually φ = 0.78 leads to only a small increase in viscosity and in and extract the width of the velocity distribution as a the width of the velocity distribution. The densest sys- function of time. These widths, ∆N vy(t), are shown as tem with φ = 0.8035 does not settle in the inertial flow a time series in fig. 15. The time series of the standard branch and the viscosity and the width of the velocity deviation reflect the overall time dependence of the sys- distribution grow as the strain rate decreases. tem as could be observed in other quantities, e.g., the shear stress which is shown for comparison. We charac- In this section, we showed that velocity fluctuations are terize the widths of the velocity distribution by fig. 16, linked to the effective viscosity of the system as also ob- where the corresponding time averaged standard devia- served in simulations of particles suspended in a viscous tions, ∆N vy(t) T , are shown. This quantity shows the fluid [28]. Both, the width of the velocity distribution same behaviorh asi the flow curve by means of the gen- and the generalized viscosity, shown in fig. 17, show the eralized viscosity, η = σ/γ˙ 2 (fig. 17): a discontinuity in same qualitative dependence on the flow state. 9

V. DISCUSSION AND CONCLUSION

We studied the shear thickening phenomenon of dry frictional granular particles. Large simulation cells exhibit continuous shear thickening across an unstable interval of stress. Continuous shear thickening is accompanied by heterogeneous and unsteady flow. The heterogeneous flow states, which were first reported in [14], possess a strong anisotropy and a characteristic length, which is considerably larger than the particle size. The characteristic length grows linearly with the system size. A length, which is based on vortexlike nonaffine displace- ments, linearly dependent on system size is reported in [29]. This is in accordance with large-scale structures that we observe by visual inspection in the velocity field, i.e., vortexlike structures and additional almost parallel moving particles (red), see fig. 19. In small simulation cells, the application of stress in the unstable interval leads to shear jamming, see also [13]. Strain rate controlled simulations show jumps across the unstable interval, see also [13, 15], giving rise to discontinuous shear thickening. The system size is found to be a relevant property for the shear thickening scenario. This FIG. 18. Snapshot of the local coordination number, Zlocal, is in stark contrast to the claim that hysteresis (accom- of the configuration in fig. 3. panied by discontinuous shear thickening) persists in the thermodynamic limit [12], where we expect continuous shear thickening instead. ber and the nonaffine velocities. This correspondence is The investigation of the nonaffine velocity distribu- shown in figs. 18 and 19, where we show the local coor- tions shows that their width is a sensible proxy for the dination number, Zlocal, and the nonaffine velocity, vna, state of the system: The time dependence of stress and respectively. In the region of large stress, the coordina- width of the velocity distribution coincide and the aver- tion number is large and in the regions of low stress the age width shows the same strain rate and packing frac- coordination number is low. Strikingly, the state, which tion dependencies as the viscosity. A recent study relates is shown, is unsteady: There is a large region of parti- velocity fluctuations to dissipation and shows that the cles (red) that moves to the bottom left with a velocity major contribution to dissipation stems from a fraction that is considerably larger than the strain rate. Regions of particles that goes to zero as jamming is approached of large nonaffine velocity correspond to regions of large [30]. We reported a novel phenomenon: a nonmonotonic stress and a low nonaffine velocity corresponds to low strain rate dependence of the width of velocity distribu- stress. An strong correlation between mean coordination tions below jamming, i.e., the fastest particles are found number and average stress was already reported [21, 31]. in the continuous shear thickening regime. It remains Here we want to emphasize the local relation. elusive how these particles contribute to the dissipation, and thereby to the rheology and the unsteady nature of flow. Appendix B: Scaling of the average standard deviation of the correlation functions

3/4 Here, we demonstrate the dependence s(N) N − . ACKNOWLEDGMENTS ∝ We consider average correlation functions with Nb bins We gratefully acknowledge ﬁnancial support by the stemming from NT samples with N particles each. Due Deutsche Forschungsgemeinschaft via programs FOR to the computational complexity and data storage, we have chosen: N 1/N . Moreover, the number of bins 1394 and Emmy Noether He 6322/1-1. MG thanks Ehsan ∝ T Irani and Knut Heidemann for fruitful discussions. is constant, Nb = const. NT snapshots contribute to the binwise standard deviation, std(C(ri)) with bin i. The number of pairs of particles per snapshot that fall in a Appendix A: Local coordination number and bin is called Nd. Therefore, we expect scaling as: nonaﬃne velocity std(C(r )) 1/ N N (B1) i ∝ T d The features that we see in the stress can be measured Nd is proportional to the size ofp the bins times the num- in other quantities, too, e.g., the local coordination number of pairs of particles, i.e., N L/N N 2 N 5/2/N . d ∝ b ∝ b 10

FIG. 19. Snapshot of the local nonaffine velocity, vna, of the configuration in fig. 3. Nonaffine velocities are indicated by arrows. The length and color scale with the absolute value of the nonaffine velocity. The reader of a digital copy is invited to zoom in and investigate the rich structure.

In total: Appendix C: Deﬁnition of the analog of the structure factor for the stress ﬁeld

1/2 5/4 3/4 std(C(r )) N /N = N − . (B2) i ∝

For all system sizes, the number of bins is constant and averaging does not aﬀect the scaling and therefore To motivate eq. 8 we consider a particle, i, at position 3/4 s(N) N − . r (t) at time t. The local stresses on the particles, σ (r; t), ∝ i i 11 constitute a ﬁeld localized at the particles’ positions: With this assumption, the Fourier transform only depends on the wave vector, k, and time:

N Cˆ(k; t) = δσ (t)δσ (t) exp [ i(k(r (t) r (t)))]. i j − i − j σ(r; t) = σ (t)δ(r r (t)). (C1) i,j i − i X i=1 (C5) X With the transformed ﬁeld

δσˆ (k; t) = δσ (t) exp ( ikr (t)), (C6) The ﬁeld, σ(r; t), has a spatial average, σ(r; t) N , and i − i δσ(r; t) = σ(r; t) σ(r; t) is the deviationh fromi the i − h iN X spatial average. Correlations in space are measure by: we can write:

Cˆ(k; t) = δσˆ (k; t)δσˆ ( k; t). (C7) − C(r, r + r0; t) = δσ(r; t)δσ(r + r0; t) (C2) The structure factor, Sσ(k; t), of a field, σ(r; t), is defined 2 = σ(r; t)σ(r + r0; t) σ(r; t) N . (C3) as the fluctuating Fourier components averaged over an − h i ensemble. Here, the ensemble that we consider is given by the time evolution of our dynamics. Thus we have with N particles: The assumption that correlations are only dependent on the displacement between two points yields: NSσ(k) = δσˆ (k; t)δσˆ ( k; t) (C8) − T D 2 E 2 = σˆ(k; t) T σˆ(k; t) T . (C9) C(r; t) = δσ(0; t)δσ(r; t). (C4) | | − |h i |

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8. Active microrheology of driven granular particles

Reprinted article with permission from Wang, Ting and Grob, Matthias and Zippelius, Annette and Sperl, Matthias Physical Review E 89 042209 (2014) http://dx.doi.org/10.1103/PhysRevE.89.042209 Copyright (2014) by the American Physical Society.

PHYSICAL REVIEW E 89, 042209 (2014)

Active microrheology of driven granular particles

Ting Wang,1 Matthias Grob,2 Annette Zippelius,2,3 and Matthias Sperl1 1Institut fur¨ Materialphysik im Weltraum, Deutsches Zentrum fur¨ Luft und Raumfahrt (DLR), 51170 Koln,¨ Germany 2Georg-August-Universitat¨ Gottingen,¨ Institut fur¨ Theoretische Physik, Friedrich-Hund-Platz 1, 37077 Gottingen,¨ Germany 3Max-Planck-Institut fur¨ Dynamik und Selbstorganisation, Am Faßberg 17, 37077 Gottingen,¨ Germany (Received 23 December 2013; published 28 April 2014)

When pulling a particle in a driven granular ﬂuid with constant force Fex, the probe particle approaches a steady-state average velocity v. This velocity and the corresponding friction coefﬁcient of the probe ζ = Fex/v are obtained within a schematic model of mode-coupling theory and compared to results from event-driven simulations. For small and moderate drag forces, the model describes the simulation results successfully for both the linear as well as the nonlinear region: The linear response regime (constant friction) for small drag forces is followed by shear thinning (decreasing friction) for moderate forces. For large forces, the model demonstrates a subsequent increasing friction in qualitative agreement with the data. The square-root increase of the friction with force found in [Fiege et al., Granul. Matter 14, 247 (2012)] is explained by a simple kinetic theory.

DOI: 10.1103/PhysRevE.89.042209 PACS number(s): 45.70.−n, 82.70.Dd, 83.60.Df, 83.10.Gr

I. INTRODUCTION regime the friction coefficients increase again and√ exhibit power-law behavior close to a square root: ζ (F ) ∝ F for Active microrheology (AM) studies the mechanical re- ex ex F 1. This finding is in contrast to the predicted constant sponse of a many-particle system on the microscopic level ex friction (second linear regime) in the colloidal hard-sphere by pulling individual particles through the system either system [8,9]. In the following, we shall demonstrate how a with constant force or at constant velocity [1]. While in schematic MCT model can capture the increase with friction passive microrheology only the linear response can be probed, for large forces. In addition, for dilute systems we shall derive AM can also be applied to explore the nonlinear response a square-root law for large forces exactly. by imposing large drag forces. An external force Fex can be imposed by magnetic [2] or optical tweezers [3]toa probe particle embedded in a soft material and then the II. DYNAMICS OF A GRANULAR INTRUDER corresponding steady-state velocity v is measured by optical The driven granular system under consideration is com- microscopy [4]. Recently, AM experiments [2] and simulations posed of N identical particles interacting with each other. One [5–7] for dense colloidal suspensions found that (i) in the probe particle experiences a constant pulling force F .The = ex linear-response region, the friction coefficient of the probe ζ dynamics of the system is given for every particle i by the Fex/ v directly indicates the increasing rigidity of the system equation of motion when approaching the glass transition from the liquid state; (ii) v =− v + i + η + in the nonlinear response region, the friction coefficient tends m˙ i ζ0 i f int i Fexδi,s, (1) to decrease to a certain value with increasing pulling force—an where ζ is the bare friction depending on the friction of the effect reminiscent of shear thinning in macrorheology. Both 0 surrounding medium, which for large values is reminiscent of effects could be explained by an extension of mode-coupling a colloidal suspension. The force f i is the particle interaction theory (MCT) to describe AM [8,9]. int force, which for the granular case is typically given by a Within the MCT interpretation, the description of AM for collision rule to include the energy loss at contact. η is a colloidal suspensions is based on the existence of a glass i random driving force satisfying a fluctuation dissipation re- transition in such systems. The interplay between growing lation η (t)η (t )=2ζ k T/mδ δ(t − t ), and the constant density correlations by glass formation and the suppression of i j 0 B i,j pulling force F is imposed on the probe particle (denoted s) those correlations by microscopic shear explains the observed ex only. Models similar to Eq. (1) have been proposed frequently behavior of the friction microscopically. In addition to col- and elaborated on regarding their dynamics in the description loidal suspensions, a glass transition is also predicted by MCT of driven granular systems [16]. for driven granular systems [10–12]. Here, the energy lost in the dissipative interparticle collisions is balanced by random agitation. Starting from the nonequilibrium steady state of A. Schematic model this homogeneously driven granular system, the corresponding The use of MCT for the description of glassy dynamics AM shall be elaborated below. can be found fully self-contained and in considerable detail In AM of granular matter similar phenomena as in colloidal in recent reviews and books such as Ref. [18]. We shall suspensions are found: (i) Dramatic increasing of the friction outline briefly the general framework of the theory as follows. coefficient and (ii) shear thinning have been identified in MCT describes the dynamics in the liquid state by correlation experiments with horizontally vibrated granular particles functions for which one can derive the microscopic equa- [13,14], and both effects were reproduced in recent simulations tions of motion for the system under consideration. Using of a two-dimensional granular system [15]. Moreover, for projection-operator formalisms, the equations of motion can large pulling forces in the simulation, beyond the thinning be reformulated exactly as integrodifferential equations for

1539-3755/2014/89(4)/042209(5) 042209-1 ©2014 American Physical Society WANG, GROB, ZIPPELIUS, AND SPERL PHYSICAL REVIEW E 89, 042209 (2014) the correlation functions where the integral terms represent liquid or glassy state together with the distance of a chosen memory effects. To close the equations, an approximation— state point from a glass-transition line, which in the case of the mode-coupling factorization—is invoked, and the resulting schematic models can be calculated analytically [18]. The = c + c + equations are solved numerically by asymptotic expansion. In state points (v1,v2) [v1(1 σ),v2(1 σ)] are specified√ by c = c c − addition to the full microscopic MCT equations, simplified a distance σ to the transition line given by v1 v2(2/ v2 1) c = so-called schematic models can be used that capture the with the specific choice v2 2 for the transition point. mathematical structure of the full theory but are easier to solve The coupling strength between the probe and the host 2 [18]. The MCT formalism has been extended to dissipative system is indicated by vA. s is the effective frequency of the 2 = − granular dynamics in Ref. [12]. The theory predicts a glass correlation function of the probe and set to be s 1 iFex. transition for a driven granular fluid for all degrees of This result can be obtained exactly from Eq. (1) by considering inelasticity, parametrized by the coefficient of resitution ε. the limit of vanishing interacting force: Hence, there is a glass transition line in the phase diagram s = − 2 2 φq (t) exp s,q t spanned by volume fraction and ε. In the following, we build (4) on these results to discuss the dynamics of a single intruder q q2 F 2 = · − → 0 − ex pulled through a dissipative granular medium. As a first step s,q (qkB T i Fex) kB T 1 i . m m q0kB T toward a full MCT theory, we generalize schematic models that have been devised for the dynamics of an intruder in a νs describes the dynamics of the density correlator of the probe | | colloidal fluid [9] to the case of an intruder inside a dissipative for short time scales. In order to assure that φs (t) 1, it is granular host fluid. required that The friction coefficient of the probe can be calculated by the νs >Fex, (5) integration-through-transients (ITT) method combined with which is obtained by solving the second equation in Eqs. (2) the MCT approximation. This procedure was first applied to without the memory kernel. By integration of the density describe the macrorheology [17] and was later extended to autocorrelators [8,9], we get the expression for the effective AM for colloidal suspension [8,9]. We follow the approach friction of the probe as in Refs. [8,9] to construct a schematic MCT model for 2 ∞ driven granular systems. Different from colloidal systems, the q0 ζ/ζ0 = 1 + kB T dt Re[φs (t)]φ(t). (6) equations of motion for the density autocorrelation functions ζ0 0 for both the bulk system and the probe particle, φq (t):= ∗ ∗ Equation (5) shows clearly that ν depends on the pulling ρ (t)ρ /ρ ρ and φs (t):=exp{iq · [r (t) − r ]},re- s q q q q q s s force. However, its dependence beyond the inequality can only spectively, include a second time derivative, because granular be obtained from a microscopic theory. Within the schematic systems are not overdamped. Here, ···denotes the ensemble model, we have investigated several functional forms of ν (F ) = N iq·r(t) s ex average and ρq (t): i=1 e is the Fourier transform complying with the constraint in Eq. (5). The simplest choice of the density. In schematic models the dependence of the is the straightforward generalization of the colloidal case: (a) correlation functions on the wave vector q is ignored. Instead, νs = 1 + Fex. As shown below, this choice is not compatible the correlation functions are evaluated at one particular with the data from simulations. This has led us to consider case wavenumber q0, e.g., the maximum of the structure factor. 2 (b) νs = 1 + F . Ultimately, a microscopic derivation needs The equations for φ(t) = φ (t) and φ (t) = φs (t) read ex q0 s q0 to show if case (b) can be obtained from the microscopic t equations of motion, cf. [9]. φ¨(t) + νφ˙(t) + 2 φ(t) + dτ m(t − τ)φ˙(τ) = 0 The respective numerical solutions for the force-dependent 0 friction coefficients are given in Fig. 1, where σ indicates t the distance from the glass transition. The force-dependent ¨ + ˙ + 2 + − ˙ = φs (t) νs φs (t) s φs (t) dτ ms (t τ)φs (τ) 0. friction of the probe exhibits three characteristic regimes. For 0 (2) small pulling forces in both models, the friction coefficient is constant, or equivalently, the average velocity of the probe is The first 3 terms in Eq. (2) describe oscillations with frequency proportional to the external pulling force. This region extends (s ), which are damped with rate ν(νs ). The last terms to external forces of order unity and describes a linear response. account for memory effects. In a schematic MCT model (called When approaching the glass transition, the friction increases F12 model) the memory kernels are approximated by nonlinear drastically as the correlation functions in Eq. (6) extend to functions of the correlation functions as follows: increasingly longer time scales. Starting around Fex ≈ 1, the 2 linear-response regime ends and gives rise to shear thinning: m(t) = v1φ(t) + v2φ (t) (3) The friction decreases and it is proportionally easier to pull the particle. Equivalently, the average velocity of the intruder ms (t) = vAφ(t)Re[φs(t)]. increases faster than linear with external force. For the model The host fluid is assumed to be large enough so that its in the left panel of Fig. 1, the friction ζ approaches the limiting density correlation function is not affected by the external value given by the bare friction ζ0 and remains there for yet pulling force Fex. Consequently, the collective dynamics is higher forces. This model hence describes behavior similar to completely decoupled in Eqs. (2) and (3). Here, we are the colloidal results for Newtonian microscopic dynamics. For interested in the long-time behavior of φ(t), which is not the model in the right panel of Fig. 1, the friction approaches affected by ν and , so that we take ν = = 1 for simplicity. a minimum around Fex ≈ 100 and starts increasing for higher The control parameters (v1,v2) of the host fluid determine the pulling forces.

042209-2 ACTIVE MICRORHEOLOGY OF DRIVEN GRANULAR PARTICLES PHYSICAL REVIEW E 89, 042209 (2014)

3 1 (a) ν = 1+F ν 2 σ = -0.02 s ex σ = -0.02 (b) s = 1+Fex Fex=1 0 0.5 ζ 2 / ] ζ s φ 10 σ = -0.1 σ = -0.1 0 Fex=250

log 1 Re[ σ = -0.5 σ = -0.5 -0.5 0 2 -11

0 Fex=250 ζ 0 σ = -0.5 σ = -0.5 0.5 V

σ = -0.1 ] F =1 -2 σ = -0.1 s ex φ σ = -0.02 σ = -0.02 0

-4 Im[ -1 0 1 2 3 4 -1 0 1 2 3 4 -0.5 log10Fex log10Fex

-2 -1 0 1 2 FIG. 1. (Color online) Force-dependent friction coefficient 10 10 10 10 10 (upper panels) for different models of the damping νs (Fex) together t with the steady-state velocities (lower panels). σ specifies the distance from the glass-transition point in the schematic model. FIG. 2. (Color online) Comparison between the schematic model and the simulation data for the density autocorrelator of the probe particle at fixed packing fraction ϕ = 0.8 and energy dissipation In comparison, the models in Figs. 1(a) and 1(b) are ε = 0.9. The pulling forces are Fex = 1 and 250. The symbols almost equivalent in the linear-response and shear-thinning represent the simulation data; the solid lines are the descriptions regimes, where the friction in Eq. (6) is dominated by the by the schematic model. memory effects leading to a slowing down of the relaxation. The difference in the microscopic damping νs does not σ =−0.13 for ε = 0.1. The other parameters are the same as play a significant role. In contrast, for large pulling forces, the ones mentioned above. the correlation function φs (t) relaxes to zero rapidly and The corresponding fit of the friction coefficients is given the integral Eq. (6) is dominated by the short-time part in Fig. 3. In the regime of small forces, the schematic model of the correlation functions. shows a linear-response plateau. The simulation data also show a plateau for small forces (see Fig. 3 in Ref. [15]). As the glass B. Comparison with simulation data transition is approached, this regime moves to smaller forces, so that it is visible in Fig. 3 only for the smallest ε = 0.1, The simulation is performed in two dimensions and the = setup is the same as described in Ref. [15]: In a bidisperse which is further away from the glass transition than ε 0.9 and 0.7. However, for ε = 0.1 and ϕ = 0.8 the simulations mixture of hard disks with size ratio Rs /Rb = 4/5 of small to big particles and a respective mass ratio m /m = 16/25, an become increasingly difficult for small forces due to the s b occurrence of long-lasting contacts. Hence, the error bars intruder of radius R0 = 2Rs and mass m0 = 4ms is suspended. All collisions are inelastic, characterized by the coefficient of become comparable to the result itself. For large pulling forces, restitution, ε. The particles are kicked randomly to balance energy input and dissipation by drag and inelastic collisions. 3 10 Lengths and masses are measured such that Rs = 1 and ms = 1 and a time scale is set by requiring that the granular temperature T = 1 in the system with F = 0. An event driven code is σ=-0.05 G ex ε=0.9 implemented to simulate N = 104 particles for a wide range of Fex and ε. ε=0.7 2 We adopt with ν = 1 + F the second schematic model to ζ s ex σ compare with the simulation data in detail. Since the schematic =-0.09 models only capture the overall mathematical structure of the 2 ε=0.1 theory, it is equally well applicable in both three and two 10 σ dimensions as the underlying glass transitions are similar in =-0.13 1/2 F both 2D and 3D [12]. The same holds for monodisperse and 1 2 3 4 bidisperse systems. 10 10 10 10 Figure 2 shows the fit of the measured correlation functions Fex by the model. The numerical solution of the density autocorrelator of the intruder fits quite well the corresponding simulation FIG. 3. (Color online) Effective friction of the intruder for differ- data for the moderately high force Fex = 250. For the smaller ent energy dissipation ε = 0.9, 0.7, and 0.1 at fixed packing fraction force Fex = 1, it shows some deviations. The fitting parameters ϕ = 0.8 of the host fluid. Individual data points show the simulation are vA = 200, σ =−0.05 for ε = 0.9 as well as (not shown results; curves represent the results from the√ schematic models. The in Fig. 2) vA = 300, σ =−0.09 for ε = 0.7 and vA = 600, dashed straight line indicates the slope of F .

042209-3 WANG, GROB, ZIPPELIUS, AND SPERL PHYSICAL REVIEW E 89, 042209 (2014) the model shows qualitatively how the increasing friction the first period: The average velocity reads coefficient can be rationalized within a schematic model. Fex − ζ0 v = − m t While the schematic model exhibits different limits for varying (t) 1 e , 0 t tc, (8) distances from the glass transition, the simulation data follow ζ0 the same curve for Fex 500. Between the extreme regimes of and the displacement of the motion satisfies large and small pulling forces the friction coefficient exhibits tc F m ζ0 ex − tc a minimum that is similar for all distances from the glass l0 = v(t)dt = tc − 1 − e m . (9) transition for both the schematic model and the simulation. 0 ζ0 ζ0 While the schematic model may only qualitatively fit the The average velocity of the probe is given by simulation data for the friction coefficient in Fig. 3, in addition tc 1 l0 to the good agreement of the correlation functions in Fig. 2, vs = v(t)dt|= . (10) the results are also consistent with the predictions from MCT tc 0 tc for hard spheres [10–12]. For smaller dissipation, i.e., larger In general, the friction of the probe vs can be calculated coefficient of restitution ε in Fig. 3, the data can be described by Eqs. (9) and (10) exactly. We first consider the two lim- m m only by choosing points closer to the glass-transition line in iting cases tc (overdamped limit) and tc (ballistic ζ0 ζ0 the schematic model. Smaller distances to the glass transition regime). given by smaller values of σ indicate that for the same density In the overdamped limit, velocity relaxation dominates over = = of ϕ 0.8 the data for ε 0.9 are much closer to the glass collisions and the collision times are large, transition than for ε = 0.1 with ε = 0.7 located in between. This finding is in agreement with the predicted increase of l0ζ0 m tc = (11) the glass-transition density with decreasing ε within MCT Fex ζ0 [10–12]. or equivalently,

Fex l0 2 . (12) III. KINETIC THEORY IN THE LOW-DENSITY LIMIT ζ0 m √ The average velocity and the friction of the intruder can To clarify the origin of the scaling law ζ ∝ Fex in the large-force asymptote, we propose a simple kinetic theory be obtained by Eq. (10) and deﬁnition of the friction itself, in the following. The simulation result from Ref. [15] has yielding shown that the scaling law is independent of packing fraction. vs =Fex/ζ0,ζ= ζ0. (13) Therefore, a potential explanation of the increased friction by jamming or shear thickening seems unlikely. Also, the The friction experienced by the intruder is dominated by the correlation functions for large pulling forces decay relatively effective friction originating from the medium. quickly, cf. Fig. 2, also contradicting a buildup of long-time In the ballistic limit, collisions dominate over velocity − ζ0 t glass-like contributions to the integrals like in Eq. (6). In the relaxation. Expanding e m in Eq. (9) to second order, we following we shall therefore focus on the low-density limit, get where exact solutions can be obtained. The formal solution of v(t)inEq.(1) can be readily obtained 2ml0 m tc = . (14) and the corresponding ensemble average of the velocity of the Fex ζ0 intruder is given by The ballistic limit is given by the presence of pulling forces very large compared to the bare friction, − ζ0 t t Fex − ζ0 e m ζ0 v = − m t + m t Fex 2l0 (t) 1 e e f int(t ) dt , (7) . (15) ζ0 m 0 2 ζ0 m The average velocity and the friction of the probe are where we have averaged out the initial velocity and the random v = η = force: 0 0 and (t) 0. l0Fex vs = ∝ Fex The direct calculation of f int(t) in Eq. (7) is difﬁcult. 2m The key point of our kinetic theory is to introduce the mean (16) free path of the intruder, l = ρ−1σ , where ρ = N/V is the 0 cr = 2mFex ∝ particle number density and σcr is the intruder’s cross section, ζ Fex. 2 l0 which for hard sphere reduces to σcr = 4πR . Let us denote the collision time as tc. Between two successive collisions Both the velocity as well as the friction are proportional to the ntc

042209-4 ACTIVE MICRORHEOLOGY OF DRIVEN GRANULAR PARTICLES PHYSICAL REVIEW E 89, 042209 (2014)

simple kinetic theory. For small and moderate external pulling forces, the schematic model agrees reasonably well with the ζ simulation data, cf. Fig. 3, and implies that the glass-transition 0=100 2 density increases with smaller coefﬁcient of restitution ε, conﬁrming predictions from mode-coupling theory [10–12]. For large forces, glassy dynamics becomes irrelevant and a ζ

10 simple kinetic theory clarifies the origin of the scaling of the ζ log ζ/ζ log =10 friction with increasing pulling force. When damping by a 0 2 10 0 1 surrounding fluid dominates the motion of the intruder at high 1 forces, a second linear regime emerges where the friction becomes constant. When collisions√ dominate, the friction 0 ∝ -2024 increases in a square-root law, ζ Fex. Such behavior of ζ2 ζ =1 log10Fex/ 0 ∝ 2 0 the drag force, Fex v , is reminiscent of Bagnold’s law for 0 -2 -1 0 1 2345 granular rheology where the shear stress is proportional to the log10Fex square of the shear rate. While the derivation of the equations above implies the FIG. 4. (Color online) Force-friction relation at large forces for existence of a thermostat to ensure a steady state at the presence different damping in the low-density limit. of energy dissipation, one may argue that for high enough velocities of the intruder, sufficient energy may be injected reason why for a driven√ granular system the friction of the into the host fluid to overcome the necessity of an additional thermostat: Consistent with that view, experiments without probe increases as ζ ∝ Fex in the large-force regime, but for colloidal hard-sphere systems the friction only decreases thermostat but high intruder velocities [19] found for high to a constant value, can be explained as follows. For different densities that the average drag force F increased quadratically 2 with the pulling velocity v of the intruder, which is consistent bare frictions ζ0,theFex-ζ plots can be rescaled as Fex/ζ √ 0 with ζ ∝ F found above. versus ζ/ζ0, cf. the inset in Fig. 4. The behavior of the probe ex in the large-pulling-force regime is determined by the ratio of In the models presented in this work, the phenomenon of the collision time scale over the Brownian velocity relaxation increasing friction ζ for high pulling forces is not specific to m the details of granular dynamics; it may only be more likely time scale, tc/ , or equivalently the value of the rescaled ζ0 to access that high-force regime in granular simulations and force F m/(ζ 2l ). In a driven granular system, the bare ex 0 0 experiments for typical control parameters. The elaboration of friction is quite small compared with the one in a Brownian the microscopic version of the theory similar to the Brownian suspension, ζ = 1 in the granular simulation [15] and ζ = 50 0 0 case [9] shall clarify this issue in the future. in the colloidal one [5]. Indeed,√ one would also obtain the same asymptotic behavior ζ ∝ Fex for Brownian systems for extremely large pulling forces. ACKNOWLEDGMENTS We acknowledge support from DAAD and DFG within IV. CONCLUSION FOR1394. We thank Andrea Fiege, Matthias Fuchs, Till We have investigated the microrheology of the driven Kranz, Leo Silbert, Thomas Voigtmann, Anoosheh Yazdi, and granular hard-sphere system by a schematic model and a Peidong Yu for valuable discussions.

[1] T. M. Squires and T. G. Mason, Annu. Rev. Fluid Mech. 42, 413 [11] M. Sperl, W. T. Kranz, and A. Zippelius, Europhys. Lett. 98, (2010). 28001 (2012). [2] P. Habdas, D. Schaar, A. C. Levitt, and E. R. Weeks, Europhys. [12] W. T. Kranz, M. Sperl, and A. Zippelius, Phys.Rev.E87, 022207 Lett. 67, 477 (2004). (2013). [3] A. Meyer, A. Marshall, B. G. Bush, and E. M. Furst, J. Rheol. [13] R. Candelier and O. Dauchot, Phys.Rev.Lett.103, 128001 50, 77 (2006). (2009). [4] J.-A. Conchello and J. W. Lichtman, Nat. Meth. 2, 920 (2005). [14] R. Candelier and O. Dauchot, Phys. Rev. E 81, 011304 (2010). [5] M. V. Gnann, I. Gazuz, A. M. Puertas, M. Fuchs, and T. [15] A. Fiege, M. Grob, and A. Zippelius, Granular Matter 14, 247 Voigtmann, Soft Matter 7, 1390 (2011). (2012). [6] D. Winter, J. Horbach, P. Virnau, and K. Binder, Phys.Rev.Lett. [16] A. Puglisi, A. Baldassarri, and V. Loreto, Phys. Rev. E 66, 108, 028303 (2012). 061305 (2002). [7] D. Winter and J. Horbach, J. Chem. Phys. 138, 12A512 (2013). [17] M. Fuchs and M. E. Cates, Phys.Rev.Lett.89, 248304 (2002). [8] I. Gazuz, A. M. Puertas, T. Voigtmann, and M. Fuchs, Phys. [18] W. Gotze,¨ Complex Dynamics of Glass-Forming Liquids— Rev. Lett. 102, 248302 (2009). A Mode-Coupling Theory (Oxford University Press, Oxford, [9] I. Gazuz and M. Fuchs, Phys.Rev.E87, 032304 (2013). 2009). [10] W. T. Kranz, M. Sperl, and A. Zippelius, Phys.Rev.Lett.104, [19] Y. Takehara, S. Fujimoto, and K. Okumura, Europhys. Lett. 92, 225701 (2010). 44003 (2010).

042209-5

Part III.

Summary and discussion

In summary, we studied rheological properties of dry granular media. In the following, we recapitulate the key results of the articles. In section 7.1, we presented a detailed study on the jamming transition of frictional granular media confined in small simulation cells. The main result is a jamming phase diagram of sheared frictional granular media based on a theory for a first-order phase transition. At finite Coulomb friction coefficient, three well separated packing fractions, (φc, φσ, φη), describe the rheological behavior:

(i) φc is the onset of rigidity, i.e., a packing fraction above which a ﬁnite stress, Σ 6= 0, can produce shear jammed states,

(ii) φσ is the lowest packing fraction where a ﬁnite yield stress, ΣY, occurs, i.e.,

for φ ≥ φσ it holds that Σ(γ ˙ → 0, φ) = ΣY(φ) > 0, and

2 (iii) φη is the packing fraction where the generalized viscosity, η = σ/γ˙ , diverges

at zero strain rate and above, for φ > φη, no inertial ﬂow is possible.

The lowest of the three packing fractions, φc, and random loose packing, φrlp, are both considered as the onset of rigidity in a frictional packing. We expect φc to equal random loose packing for inﬁnite Coulomb friction parameter [Silbert,

2010]. The fact that we can assign a physical meaning to φσ and φη, is an advancement compared to treating them as “fictitious” or “fitting” parameters for scaling laws Otsuki and Hayakawa [2011] (therein called φS and φL, respectively). Moreover, we observe reentrant flow: Starting from a jammed configuration at finite stress, and then lowering or increasing the imposed stress, leads to flow. Instead of the inertial flow phase at low stress in our reentrance topology, in Bi et al. [2011], the authors report fragile states, i.e., anisotropic jammed states with force chains, which percolate in only one direction. The difference between the inertial flow phase at low stress and the fragile states stems from the fact that our grains are embedded in a frictionless environment, which allows for ballistic motion and thus inertial flow. In the experiment by Bi et al. [2011], particles, which are subject to quasistatic shear, move on a bottom plate with a finite roughness, i.e., no inertial flow is possible. Fragile states are eliminated when friction with the bottom plate is ruled out [Zhang et al., 2015]. Another route to fragile states is cohesion between particles [Zhang et al., 2015; Irani et al., 2014, 2016]. Our simulations neither incorporate friction with a bottom plate nor cohesion, consistent with the absence of fragile states.

75 In section 7.2, we studied the rheological response of a frictional granular medium in a large simulation cell. The main result is the existence of strongly time-dependent and heterogeneous states. We showed the existence of unsteady states by prescribing stress in the unstable region of a flow curve. The hydrodynamic model, which is derived in this work, couples hydrodynamics to the evolution of a microstructure variable, which we associate with contributions to the dynamics that stem from friction. The steady state solutions of the hydrodynamic model corresponding to flow equal the flow curves in the previous article, see Grob et al. [2014] and section 7.1. The steady states are investigated with linear stability analysis that is in agreement with the simulation.

In the phase diagram for large granular systems [Grob et al., 2016], three packing fractions, (φc, φσ, φη), are necessary to describe the rheology — just as for small simulation cells, as studied in [Grob et al., 2014]. From the discussion of the phase diagram and the previous study, [Grob et al., 2014], we expect that without the contribution associated with friction, the frictionless scenario is recovered and only steady and homogeneous flow is possible. Unsteady states, either oscillatory or chaotic, have been shown to exist in scalar models [Cates et al., 2002]. Moreover, experimental findings show unpredictable flow behavior in a potential discontinuous shear thickening regime [Hermes et al., 2015]. However, this work presents the first example of chaotic flow for a dry granular medium.

Section 7.3 dealt with the characterization of ﬂow states of dense frictional granular matter. It connects the previous studies Grob et al. [2014] and Grob et al. [2016]. In both previous studies, we presented a phase diagram. These phase diagrams show an identical sequence of characteristic packing fractions,

(φc, φσ, φη). However, the rheology of the frictional granular medium, and thus the phases in these phase diagrams, differ between the two studies. We showed that this difference stems from heterogeneities. The size of the simulation cell allows granular media in only large systems to develop heterogeneities in the unstable region of a flow curve. Heterogeneities imply unsteady flow, which leads to shear thickening in the flow curve. Small systems instead show discontinuous shear thickening in a strain rate controlled setting but jam when driven by an imposed stress in the unstable region of a flow curve. We show that the global stress is related to the width of the nonaffine velocity distribution. This width is maximal in the heterogeneous flow state in which the generalized viscosity is also

76 maximal. Stress and the width of the nonaffine velocity distribution follow the same time dependence. Moreover, we report strong spatial correlations between stress and nonaffine velocities. In particular, the strong heterogeneities in the stress fields are accompanied by large gradients in the nonaffine velocity fields. In addition to the stress, which is measured in an experiment or a simulation, the nonaffine velocity is a key quantity since dissipative contributions to particle interactions are typically velocity dependent. In Olsson [2016], the author points out how rheology is governed by dissipation via the energy balance. In the frictionless system in [Olsson, 2016], the width of the velocity distribution increases monotonically as jamming is approached. This increasing width means that the number of particles dominating dissipation decreases. These particles are the fastest particles. Below jamming, we report the fastest particles, which presum- ably contribute most to dissipation, in the shear thickening regime. Both, flow in the proximity of jamming and shear thickening, are flow regimes that possess a broad nonaffine velocity distribution and, consistently, both are regimes that are governed by heterogeneity and cooperative motion giving rise to enhanced velocities. In Chapter 8, the dynamics of an active probe particle near the granular glass transition was probed by numerical simulations and related to analytical results. The steady state behavior of the probe particle provides a velocity-force relation. This velocity-force relation is linear for low forces far from the glass transition. Close to the glass transition, the linear response regime is followed by a superlinear velocity-force dependence. For large forces, we report a sublinear velocity-force relation, independent of the packing fraction. The kinetic theory by Wang and Sperl in Wang et al. [2014] rationalized the sublinear dependence in the granular system. Experiments show [Habdas et al., 2004] that also colloids display a superlinear velocity-force dependence close to the glass transition. This effect is rationalized by a localization/delocalization transition that takes place when a packing fraction dependent force threshold is crossed [Gazuz and Fuchs, 2013]. The sublinear force thickening regime at large forces, which we observe for granular particles, is in contrast to linear or superlinear regimes in colloidal systems, as observed in experiments [Habdas et al., 2004; Wilson et al., 2009], numerical simulations [Carpen and Brady, 2005], and predicted by mode coupling theory [Gazuz et al.,

77 2009]. The kinetic theory by Wang and Sperl [Wang et al., 2014] also links the microrheology studies and the shear rheology studies together. The theory states that in a collision dominated regime, the friction coefficient grows as a square- 1/2 root with the external force: ζ ∝ Fext . This implies that the velocity grows 1/2 2 as v ∝ Fext or v ∝ Fext — analogous to Bagnold’s results for shear rheology studies, where the only relevant time scale of the inertial flow regime is set by the strain rate [Bagnold, 1954]. In both cases, the inertial flow in shear rheology and the limit of large driving forces in microrheology, the dynamics is governed by collisions of hard grains. Therefore, the only relevant time scale is the external time scale, i.e., γ˙ in shear rheology or Fext in microrheology. The other flow regimes of the shear- and microrheology studies cannot be connected directly. The shear rheology studies are carried out with particles with a finite elasticity, which sets a time scale and allows for plastic flow. This timescale is absent in the microrheology studies since the particles are infinitely hard. If they possessed a large (but finite) stiffness the collisional regime would be followed by elastic-inertial flow at a driving force once particle deformations would become important. The stochastic agitation of the bath particles in the microrheology studies sets a time scale that leads to an effective granular temperature and thus diffusion. This time scale is absent in our shear rheology studies, which exclude flow regimes other than quasistatic or inertial flow at low strain rate. In essence, this thesis provided (i) a thorough study of the jamming phenomenon of dry frictional particles and (ii) the examination of the dynamics of a probe particle in a granular medium close to the granular glass transition. Both, shear rheology and microrheology, treated transition phenomena of nonequilibrium systems. In the shear rheology study, we presented results that rely on the presence of friction. The incorporation of friction increases the complexity of the jamming scenario but provides a stronger connection to experimental results. We provide a shear thickening and shear jamming scenario of a dry granular system and report, for the first time, unsteady flow states. The microrheology studies revealed the role of the different time scales whose interplay determines the dynamics of a probe particle by which the rheological properties of its surrounding are characterized. Overall, this thesis sheds light on fundamental aspects of the peculiarities of particulate media rheology.

78 Part IV.

Outlook

The work presented in this thesis contributes to a better understanding of the ﬂow of dry frictional particles under simple shear. Three major future directions are directly implied by this work:

(i) the investigation of the microscopic processes,

(ii) the characterization of the chaotic nature of the unsteady ﬂow,

(iii) and the study of frictional particles in Poiseuille ﬂow geometry.

In the following, we discuss the directions (i)-(iii) in detail. On a microscopic scale, friction opens two new channels of energy dissipation: in addition to normal viscous damping, particles can rotate and dissipate energy during sliding or non-sliding interactions. The role played by different dissipation mechanisms is crucial for the flow of granular media [DeGiuli et al., 2015]. As show in this thesis, the transition from flowing states into jammed states is also highly affected by friction. When a frictional granular medium jams, all the contacts become non-sliding since sliding is a dynamic phenomenon. For infinite friction coefficient, however, all contacts are non-sliding, irrespective of the state of the granular medium. In contrast, in the frictionless limit, all contacts slide but jamming at finite Σ below φJ is not possible. The role of sliding contacts and dissipation via sliding in the (shear) jamming picture remains elusive. Moreover, the relation between sliding and shear transformations — plastic, dissipative, and irreversible events — is not yet studied. It is challenging to detect the chaotic nature of the unsteady flow. Several algorithms exist that allow for the computation of Lyapunov exponents of a dynamical system based on a time series of a signal, see, e.g., [Sano and Sawada, 1985; Geist et al., 1990; Rosenstein et al., 1993; Kantz, 1994; Small et al., 2013]. For an accurate estimate of the Lyapunov exponent the time series needs to be sufficiently long but with high resolution. In addition, a zoo of (technical) parameters needs to be tuned according to the dynamics. The presence of mechanical noise, which stems from the collisions of grains, makes a distinction between mechanical noise and chaotic dynamics a delicate problem. An unusual path in studying the jamming transition is taken in Banigan et al. [2013]. There, the authors apply techniques of nonlinear dynamics to molecular dynamics simulations close to the jamming transition and report not only a transition from liquid to solid state near φJ but also from chaotic to non-chaotic dynamics when the strain

81 ϕ=0.8035, N=8000, p=1x10-8 ϕ=0.8035, N=8000, p=1x10-4 1 0.8 0.6 L y y/L 0.4 x 0.2 0 -2 0 2 4 6 8 10 12 14 1/2 vx / p

Figure 8.1.: Inset: Poiseuille flow geometry with suggested parabolic velocity profile. Main: Non-parabolic profile of streaming velocity, vx, showing flow (large pressure, green) and arrest (low pressure, purple). Units according to [Grob et al., 2014]. rate of the system is fixed. The dependence of the largest Lyapunov exponent on the packing fraction suggests that a non-chaotic regime exists at low packing fractions far from the jamming transition, i.e., in the inertial flow regime. Poiseuille flow, i.e., the flow through a pipe, is a natural step in the direction of the investigation of granular media under shear. Simple shear has the conceptual advantage that hydrodynamics implies homogeneous strain rate and stress profiles in a steady state. In contrast, a Newtonian fluid flowing through a pipe shows a non-homogeneous steady state: The velocity profile is a parabola, see inset of figure 8.1. Therefore, we expect that the strain rate profile and the stress profile in a granular fluid are non-homogeneous, too. Figure 8.1 shows a set of velocity profiles that show non-parabolic velocity fields and a clear distinction between flowing and non-flowing states. These phenomena have already been reported in studies of particulate media, see, e.g., [Varnik and Raabe, 2008; Chikkadi

82 and Alam, 2009; Chaudhuri and Horbach, 2013], but frictional effects have not yet been taken into account. In simple shear geometry, steady state flow is homogeneous and probes one point in our proposed phase diagrams. In contrast, Poiseuille flow is inherently heterogeneous and samples not only a point in the phase diagram but a line, e.g., at constant packing fraction but ranging over an interval of stress. It would be interesting to study whether Poiseuille flow exhibit several coexisting phases with different stress at the same time, i.e., blobs of shear jammed particles and areas of a flowing phase, either in plastic or inertial flow.

Part V.

Appendix

A. Details on the hydrodynamic model

In this Chapter, we detail the stability analysis, of the model that we present in the article “Rheological chaos of frictional grains” in Chapter 7.2. Here, we use the second equation in equations 2 in [Grob et al., 2016],

γ˙ (σ, w) =γ ˙ 0(σ) − w, (A.1) and formulate the model, equations 2 from Grob et al. [2016], via two equations:

2 ∂t(γ ˙ 0(σ) − w) = ∂y σ γ˙ ∂ w = − (w − w∗). (A.2) t Γ

The microstructure variable, w, evolves towards a stress dependent value: w∗ = w∗(σ). As w describes frictional contributions, w∗ should be zero at zero stress and it should grow with the stress that is applied. The simplest relation which fulﬁlls these requirements is w∗(σ) = bσ. The model allows for two stationary states, i.e., the left hand sides of equations

A.2 are set to zero. In a stationary state, (σ0, w0), the stress is homogeneous and w determines the strain rate via equation A.1. The ﬁrst solution corresponds to stationary ﬂow: ∗ w0 = w (σ); σ0 = σ. (A.3)

The ﬂowing steady state solution A.3 recovers the phenomenological constitutive equation, equation 1 in article Grob et al. [2014], via equation A.1. This implies for the slope

∂σγ˙ |σ0 = ∂σγ˙ 0|σ0 − b. (A.4)

87 A. Details on the hydrodynamic model

The second stationary solution accounts for the shear jammed state:

w0 =γ ˙ 0(σ0); σ0 = σ. (A.5)

In this solution, inserting equation A.5 in equation A.1 leads to zero strain rate. In the following, we discuss the stability of the solutions in equations A.3 and A.5. We employ linear stability analysis and consider small deviations, δσ = σ −

σ0 and δw = w −w0, from the steady state solution, (σ0, w0), of the equation A.2.

A.1. The ﬂowing state

The linearization of the coupled equations A.2 requires the linearization of each term. To linearize around the ﬂowing solution, equation A.3, we linearize the frictionless constitutive relation, γ˙ 0(σ), and its derivative: γ˙ 0(σ0 + δσ) =γ ˙ 0(σ0) + 1 −1/2 2 1 −3/2 2 ( 2 aσ +2cσ)δσ +O(δσ ) and ∂σγ˙ 0(σ) = ∂σγ˙ 0(σ0)−( 4 aσ0 +2c)δσ +O(δσ ), respectively. Up to linear order in the small quantities, δσ and δw, or products of these, and with the notation τ = Γ/γ˙ (σ0) equation A.2 reads:

∂yyδσ − ∂σγ˙ 0|σ0 ∂tδσ + ∂tδw = 0 b 1 δσ − δw − ∂ δw = 0. (A.6) τ τ t

By Fourier transform, we substitute ∂y → ik, with wave number k, and ∂t → iω := Ω, with frequency, ω or growth rate Ω. The Fourier transform of equations A.6 can be written in matrix notation:       2 ˆ 0 −k − ∂σγ˙ 0|σ0 ΩΩ δσ   =     . (A.7) b 1 ˆ 0 τ − τ − Ω δw

The hat indicates the transformed variables. If the determinant of the matrix in equation A.7 vanishes there can be more than the trivial solution (δσ = 0 = δw). Hence, we have

2 2 1 2 k 0 = Ω + Ω (τk + ∂σγ˙ 0|σ0 − b) + . (A.8) τ∂σγ˙ 0|σ0 τ∂σγ˙ 0|σ0

The polynomial is solved for its roots given all the constants. The roots lead to

88 A.2. The shear jammed state the dispersion relation Ω(k) = Ω0 + iΩ00 with

A finite coulomb friction parameter, µ > 0, i.e., b > 0, is necessary for the growth of perturbations and oscillations since only then the growth rate, Re(Ω), can become positive due to a negative slope, ∂σγ˙ |σ0 , of the flow curve. In the frictionless case, i.e., when b = 0, the dispersion relation is always negative giving rise to a negative growth rate and stable flow. When τ gets small, i.e., infinitely fast relaxation of w to w?(σ), or k → 0, i.e., infinitely large systems, the dispersion relation becomes proportional to the slope of the frictional flow curve, which makes the frictional contribution to the slope, −b, become increasingly important.

A.2. The shear jammed state

A linearization of the coupled equations A.2 around the shear jammed solution, equation A.5, yields up to linear order

∂yyδσ − ∂σγ˙ 0|σ0 ∂tδσ + ∂tδw = 0 1 − (w − bσ )(∂ γ˙ | δσ − δw) − ∂ δw = 0. (A.10) Γ 0 0 σ 0 σ0 t

The Fourier transformed set of equations A.10 can be written in matrix notation:       2 ˆ 0 −k − ∂σγ˙ 0|σ0 ΩΩ δσ   =     . (A.11) Γ ˆ 0 ∂σγ˙ 0|σ −1 + Ω δw 0 w0−bσ0 As above, if the determinant of the matrix in equation A.11 vanishes there can be more than the trivial solution (δσ = 0 = δw). Hence, we have

1 w − bσ 0 = Ω2 + Ωk2 − k2 0 0 . (A.12) ∂σγ˙ 0|σ0 Γ∂σγ˙ 0|σ0

The solutions of equation A.12 lead to the dispersion relation of the shear jammed

89 A. Details on the hydrodynamic model state: v 2 u 2 !2 1 k u 1 k 2 w0 − bσ0 Ω1/2 = − ± t + k . (A.13) 2 ∂σγ˙ 0|σ0 2 ∂σγ˙ 0|σ0 Γ∂σγ˙ 0|σ0

Below φσ, see [Grob et al., 2014, 2016], there is potentially one unstable and one stable mode when w0 − bσ0 is positive and large enough, i.e., Re(Ω1) > 0 and Re(Ω2) < 0. Shear jamming is then unstable and a flowing solution is attained. Above φσ, the strain rate gets negative and both solutions are stable when w0 −bσ0 < 0. This is what we identify with shear jamming. Note that here, in contrast to the flowing state, the parameter b enters via the full flow curve, w0 − bσ0 =γ ˙ 0(σ0) − bσ0, and not via the slope.

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104 Author contribution

Jamming of frictional particles: A nonequilibium ﬁrst-order phase transition

Grob, M., Heussinger, C., and Zippelius, A. Jamming of frictional particles: a nonequilibrium ﬁrst-order phase transition. Physical Review E 89:050201 (2014). http://dx.doi.org/10.1103/PhysRevE.89.050201 The article covers molecular dynamics studies of frictional grains and the analysis of a phenomenological model. I developed the code of the simulation, conducted all the numerical simulations, analyzed the model, and created all the ﬁgures and the supplemental material. The study was designed by me, Claus Heussinger, and Annette Zippelius. The article was mainly written by Annette Zippelius.

Rheological chaos of frictional grains

Grob, M., Zippelius, A., and Heussinger, C. Rheological chaos of frictional grains. Physical Review E 93:030901 (2016). http://dx.doi.org/10.1103/PhysRevE.93.030901 The article covers molecular dynamics studies of frictional grains and the analysis of a hydrodynamic model. I developed the code of the simulation, conducted all the numerical simulations, derived and analyzed the model, and created all the ﬁgures and the supplemental material. The study was designed by me, Claus Heussinger, and Annette Zippelius. The articles was written by me, Claus Heussinger, and Annette Zippelius.

105 Author contribution

Unsteady rheology and heterogeneous ﬂow of dry frictional grains

The manuscript covers molecular dynamics studies of frictional grains. I developed the code of the simulation, conducted all the numerical simulations, and created all the ﬁgures. The study was designed by me, Claus Heussinger, and Annette Zippelius. The manuscript was written by me with revisions from Claus Heussinger and Annette Zippelius.

Active microrheology of driven granular particles

Wang, T., Grob, M., Zippelius, A., and Sperl, M. Active microrheology of driven granular particles. Physical Review E 89:042209 (2014). http://dx.doi.org/10.1103/PhysRevE.89.042209 The article covers a study that was designed by Ting Wang and Matthias Sperl. I contributed the molecular dynamics results. Ting Wang and Matthias Sperl wrote the article with support by me and revisions from Annette Zippelius.

106 Acknowledgments

Meiner Betreuerin Prof. Dr. Annette Zippelius möchte ich einen tiefen Aus- druck von Dank entgegenbringen. Sie hat mich während der letzten Jahre stets gefördert und gefordert, viele interessante Dienstreisen wahrnehmen lassen und war stets bereit Ratschläge und Meinungen in sämtlichen Belangen zu äußern. Herrn Dr. Claus Heussinger, ebenfalls mein Betreuer, gilt auch großer Dank. Er hat produktive Denkanstöße gegeben und war für inhaltliche und auch technische Probleme immer ein hilfreicher Ansprechpartner. Prof. Dr. Reiner Kree danke ich für die Übernahme des Zweitgutachtens. Weiter möchte ich den Mitarbeitern der Gruppen Zippelius und Heussinger danken. Allen voran Elham und Ehsan, Till, Florian, Ronny, Andrea und Iraj. Special thanks go to the group of Prof. R.P. Behringer at Duke University, Durham, NC, USA. He and his (former) group members Jonathan Barès and Dong Wang gave me hospitality in the probably most-established laboratory for granular materials. Großer Dank gilt auch der Verwaltung des Instituts. Durch Frau Schubert und Frau Glormann und meine Rolle als Gleichstellungsbeauftragter habe ich viel über administrative Vorgänge im Institut lernen können. Technische Unterstützung kam vor allem von Dr. Jürgen Holm und Thomas Köhler, die die Hard- und Software im Institut pﬂegen. Die vielen CUDA-Rechner wären ohne Andre Galuschko nicht ins Institut gekommen. Weiter möchte ich mich bei Freunden und Familie bedanken. Allen voran geht hier Knut Heidemann, mit dem ich seit dem Studium freundschaftliche und fachliche Kontakte pﬂege. Besonders bemerkenswert ist sein Engagement vor der Fertigstellung meiner Dissertation. Meiner lieben Friederike möchte ich danken, dass sie des Öfteren mit lieben Worten und Taten für Ausgleich zur Arbeit sorgen konnte. Japser Cirkel möchte ich für weiteres Korrekturlesen danken. Elterliche Unterstützung hätte ich nicht missen wollen.

107

List of Publications

Publications related to this thesis

• Grob, M., Zippelius, A., and Heussinger, C. Rheological chaos of frictional grains. Physical Review E 93:030901 (2016). http://dx.doi.org/10.1103/PhysRevE.93.030901

• Grob, M., Heussinger, C., and Zippelius, A. Jamming of frictional particles: a nonequilibrium ﬁrst-order phase transition. Physical Review E 89:050201 (2014). http://dx.doi.org/10.1103/PhysRevE.89.050201

• Wang, T., Grob, M., Zippelius, A., and Sperl, M. Active microrheology of driven granular particles. Physical Review E 89:042209 (2014). http://dx.doi.org/10.1103/PhysRevE.89.042209

Further publications

• Fiege, A., Grob, M., and Zippelius, A. Dynamics of an intruder in dense granular ﬂuids. Granular Matter 14:247 (2012). http://dx.doi.org/10.1007/s10035-011-0309-9

109